m2 atoms and photons complete lecture notes

Atoms and photons

J.M. RaimondUniversite Pierre et Marie Curie

Laboratoire Kastler BrosselDepartement de Physique de l’Ecole Normale Superieure

[email protected]

September 6, 2016

Table des Matieres

1 Classical and phenomenological approach 91.1 A classical model: the harmonically bound electron . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Diffusion by an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.4 Propagation in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Einstein’s coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 The three coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Relations between the coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.3 Stimulated emission: the laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.4 A simple model of laser radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.5 Properties of laser radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Semi-classical approach 192.1 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Gauge choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 A ·P interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.3 D ·E interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Non-resonant interaction: Perturbative approach . . . . . . . . . . . . . . . . . . . . . 232.3 Free atom and resonant field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Atomic system and interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Rabi precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.3 Ramsey separated oscillatory fields method . . . . . . . . . . . . . . . . . . . . 32

2.4 Relaxing atom and resonant field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1 Description of the atomic relaxation . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2 Quantum jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3 Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5.1 The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.2 Two limit cases: back to Einstein’s coefficients . . . . . . . . . . . . . . . . . . 45

2.6 Applications of the optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . 502.6.1 Steady-state and saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.2 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.6.3 Dark resonances and EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6.4 Maxwell-Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Field quantization 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1.1 Planck 1900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1.2 Einstein 1905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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4 TABLE DES MATIERES

3.2 Field eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.1 Mode decomposition of the electromagnetic field . . . . . . . . . . . . . . . . . 673.2.2 Normal variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2.3 Field energy: canonical variables . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.4 Field momentum and angular momentum . . . . . . . . . . . . . . . . . . . . . 74

3.3 Field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.1 Creation and annihilation operators, Hamiltonian . . . . . . . . . . . . . . . . . 763.3.2 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Field quantum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4.1 Fock states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4.2 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4.3 Phase space representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5 Coupling field modes: the beamsplitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.5.1 Hamiltonian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.5.2 State transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.6 Field relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.6.1 Lindblad equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.6.2 Evolution of some states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Coupling quantum field to matter 1094.1 Interaction Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1.1 Quantum field and classical charges . . . . . . . . . . . . . . . . . . . . . . . . 1094.1.2 Quantum field and quantized atom . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 Spontaneous emission in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2.1 Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2.2 Wigner-Weisskopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.3 Photodetection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.1 Photodetector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.2 Intensity correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4 The dressed atom model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.4.1 The atom-field eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.4.2 Resonant case: Rabi oscillation induced by n photons . . . . . . . . . . . . . . 1254.4.3 Non-resonant coupling and single-atom index effects . . . . . . . . . . . . . . . 1274.4.4 Two applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5 Cavity Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Introduction

The atom-field interaction is certainly one of the most important problems in quantum physics at lowenergies. Most of the information we get on the visible universe, most of the actions we may haveon it are mediated by the interactions of the atoms with the electromagnetic field. The quantumtheory of these interactions, the “quantum electrodynamics” is one of the key successes of last centuryphysics. It provides an extremely detailed insight, with an impressive compatibility with the availableexperimental data. For instance, the spectrum of the simplest atom, hydrogen, can be computedand compared with experiments with a precision at the 10−12 level, only limited by our imperfectknowledge of the structure of the proton. Quantum optics, the manipulation of fields with a few lightquanta only, has also led to very impressive vindications of the most intimate aspects of quantumphysics, for instance with the direct evidence of quantum non-locality in the famous experiments ledby A. Aspect in the 80s.

The matter-field interaction at the quantum level has also led to a flurry of important practicalapplications. The laser, at the heart of the modern information society (any message on the internetat one time transits as a collection of light pulses in an optical fibre), is a direct application ofour understanding of the atom-field coupling. The atomic clocks are the heart of the GPS system.Their precision is considerably improved by the cold atom and trapped ions techniques, leading to animpressive stability (in the 10−17 range). The cold atoms science reflects our ability to use light tochange the motion of atoms. It culminates in the Bose Einstein condensation, observed for the first timein 1995. This new state of matter can be used as a fully controllable prototype of a condensed mattersystem. The addition of a periodic potential created by a light standing wave realizes a ‘quantumsimulator’ of various properties (quantum phase transitions, topological phases...) too complex to becomputed or modelled accurately.

These lectures will not of course cover all aspects of an extremely rich problem. We will focus onthe basic aspects of quantum light-matter interaction. How does a single or a few atoms interact withnear-resonant light? We will pass progressively from the simplest classical or qualitative approachesto the final explicit treatment of the interaction of a quantum atom with a quantum field.

The first Chapter will be devoted to the interaction of atoms, classically modelled as a chargedharmonic oscillator, with a classical field. This model is of course extraordinarily naive, but we will seein the following Chapters that its conclusions are surprisingly close to those of fuller quantum models.We will also introduce in this Chapter the Einstein coefficients describing phenomenologically therate of energy exchange between the atoms and the field. The consistency of this model requires theintroduction of a stimulated emission process, which is at the heart of the laser (or maser) principle.We will later recover these coefficients in a more rigorous way and get insight into their validity domain.

The second Chapter will be devoted to the interaction of a quantized atom with a classical field.When this field is far off-resonance from any atomic transition, we will recover, mutatis mutandis, thebasic conclusions of the charged harmonic oscillator model. At resonance, we will describe the newand important phenomenon of Rabi rotation, and discuss a few of its applications such as Ramseyinterferometry, a key component of high-resolution spectral measurements. This first discussion willassume that the atom is, besides its interaction with the field, isolated from the outside world. Thisis never the case in practice, be it only due to the spontaneous emission phenomenon, which makes

5


all atomic excited states unstable (the rate of this emission will be precisely computed in Chapter 4by a fully quantum approach).

We will thus introduce in Chapter 2 the appropriate tools for the description of quantum relaxation,or decoherence, i.e. the coupling of a quantum system to a complex reservoir at thermal equilibrium.We will do so in a framework (Kraus operators) inspired by the recent developments of quantuminformation theory and providing a very clear and intuitive picture of the relaxation phenomenon.We will later show how this approach leads to the standard master equation, describing the timeevolution of the reduced density matrix of the atom. Equipped with these equations, we will beready to treat the interaction of a relaxing atom with a classical field, and to establish the so-calledoptical Bloch equations. We will discuss a variety of applications of these equations, from high-resolution spectroscopic methods based on the saturated absorption, to electromagnetically inducedtransparency (EIT) and slow light.

This semi-classical approach, if extremely useful, does not tell the whole story. We now need toproperly quantize the electromagnetic field. We will do so in Chapter 3. The first step is to finda convenient mode basis for the classical field, solution of the Maxwell equations with appropriatelimiting conditions. We will then write the Hamiltonian of these classical modes and identify themwith those of 1-dimensional harmonic oscillators. The canonical quantization of the field will then bea simple problem. We will introduce the field operators, the field energy eigenstates and the coherentstates, which are the closest possible analogue of a classical field. Finally we will introduce a fewpictorial representations of these states in the phase plane (the so-called Fresnel plane). We will alsotreat the coupling of two field modes by a semi-transparent mirror (a beam-splitter) and recognizethat this simple device produces extraordinarily complex entangled quantum states. The final part ofthis Chapter will be devoted to the exploration of field decoherence. We will in particular show thatthe coherent states are particularly robust with respect to decoherence.

The final Chapter will be devoted to the interaction of a quantum field with a quantum atom,completing our initial goals. We will begin with the spontaneous emission problem and show howits rate can be precisely computed. We will also guess that the coupling of the many non-resonantfield modes with the atom produces a shift of the atomic levels (the Lamb-shift). The computationof this shift is a difficult problem, that jeopardized for a long time the development of quantumelectrodynamics. Any simple model leads to an infinite shift value (and there are many other infinitiesin simple-minded QED). The regularization of these infinities, by the techniques of renormalization,is clearly beyond the scope of these lectures. Since we will focus mainly on the interaction of a fewresonant modes (laser, cavity modes) with the atom, we can indeed forget these shifts or treat themphenomenologically as a slight modification of the bare atomic levels.

We will thus turn to the central part of this Chapter, the interaction of a single nearly resonantmode with a quantum two-level atom. This situation, initially explored by Jaynes and Cummings,is a rather approximate approach to the laser-atom interaction in free space. On the contrary, itis a perfectly precise description of the Cavity Quantum Electrodynamics situation, describing thecoupling of a single atom with a field contained in a mode of a high quality cavity. This domain,which originates theoretically in the 40s with a seminal remark by Purcell, has and is still undergoinga remarkable development, recognised by the Nobel prize in Physics awarded in 2012 to Serge Haroche,one of the leading pioneers of CQED.

We will thus devote a rather large part of this final Chapter to CQED, first because it is a directapplication of the formalism developed in these lectures, but also because it leads to many interestingquantum situations, including the realization of some of the gedankenexperiments used by the foundingfathers to assess their interpretation of the formalism.

These notes will not cover all the topics related to the atom-photon interaction. There are manyexcellent textbooks in the field. Let us quote Cohen-Tannoudji, Dupont-Roc and Grynberg, Anintroduction to quantum electrodynamics and Photons and atoms, Wiley, 1992; Cohen-Tannoudji andGuery Odelin Advances in atomic physics: an overview, World Scientific 2012 (these three volumes

TABLE DES MATIERES 7

cover all the material of these lectures), but also Schleich, Quantum optics in phase space, Wiley 2000;Vogel, Welsch and Wallentowitz, Quantum optics an introduction, Wiley 2001; Meystre and SargentElements of quantum optics, Springer 1999; Barnett and Radmore Methods in theoretical quantumoptics, OUP, 1997; Scully and Zubairy Quantum optics, 1997; Loudon Quantum theory of light, OUP1983. A part of this document has bee extracted from Haroche and Raimond Exploring the quantum,OUP 2006, which covers in details all the cavity QED final part.

Last but not least, these lecture notes are in a large part directly translated from the French versionwritten for this same course by Claude Fabre a couple of years ago.

Chapitre 1

Classical and phenomenologicalapproach

In this Chapter, we present two simple approaches to the atom-fied interaction. First, we will describethe interaction of a classical field with an harmonically bound charge. This is admittedly a rathernaive model of an atom (often called Thomson model, and proposed at the end of the 19th centurywhen the existence of the electon was first revealed). However, we will see later that it captures manyessential aspects of the interaction and that its predictions coincide rather well with those of a morerefined semi-classical model.

The second part of the Chapter will be devoted to the Einstein coefficients, describing in a heuristicway the energy exchanges between matter and field. They are of course essential for the description ofa laser system. We will show later that the semi-classical model based on the Optical Bloch Equationsindeed coincides, in the proper limits, with this simple model.

1.1 A classical model: the harmonically bound electron

1.1.1 Equations of motion

We consider thus an atom as made of a single electron (negative charge q = −e, mass m, bound tothe origin (we assume the atom motionless) by a harmonic force1. The position of the electron beingr, the equation of motion is:

d2r

dt2+ ω2

0r = 0 , (1.1)

where ω0 is the atomic resonance frequency (about 1015 Hz). The solution of this equation is obviouslyr = r0 exp(−iω0t). This is not a very physical solution. We should include some damping reflecting thefinite lifetime of the excited states. We will consider here only the radiative damping, correspondingto the emission of light by the accelerated, radiating electron. We will thus be able later to comparethe predictions of this simple classical model to those of the full quantum approach.

The interplay between the electronic motion and the emission of radiation by the electron is apriori complex. We can simplify it a bit by noting that the radiation damping of the electronic motionis a weak perturbation. The typical lifetime of an atomic excited state is in the 10−8 s range, and thequality factor of the motion is thus about 107. We can thus use the instantaneous power radiated bythe electron, given by the Larmor formula directly deduced from the Lienart-Wichert potentials (and

1This harmonic force derives from an early atomic model, Thomson’s “plum pudding”, describing the motion ofelectrons – the only known atomic component at the time – in the electric field of a uniformly positively charged sphere.

9

10 CHAPITRE 1. CLASSICAL AND PHENOMENOLOGICAL APPROACH

assuming implicitly that the motion is not strongly affected by the radiation):

P =q2a2

6πε0c3= mτa2 , (1.2)

where a is the electron acceleration and where we introduce the characteristic time τ given by:

τ =1

6πε0

q2

mc3= 6.32 10−24 s . (1.3)

This time is simply related to the classical radius of the electron2

re =q2

4πε0mc2= 3. 10−15 m (1.4)

by

re =3

2cτ . (1.5)

The time τ is thus of the order of the time needed by light to cross an electron!In order to include radiation damping in the equations on motion, we want to get an effective

force acting on the electron. We will thus write the radiated power as that of a force. Integrating thepower over a time interval [t1, t2] long at the scale of τ and long compared to the electron oscillationfrequency but short on the time scale of energy dissipation, we get the energy variation as

∆E = mτ

∫ t2

t1a2 dt = −mτ

∫ t2

t1

d2v

dt2· v dt+mτ [a · v]t2t1 , (1.6)

where we have performed an integration per parts. For a bounded motion, the integral term overa time long compared to the period overwhelms the last integrated term on the r.h.s. We can thusneglect this last term and we get directly the radiated power as the opposite of that of a ‘radiationreaction force, fr:

fr = mτda

dt. (1.7)

This radiation reaction force must be taken with a grain of salt, since it does not enter in thestandard Newtonian framework: it involves the derivative of the acceleration, meaning that the initialacceleration is an important inital condition for solving the equations of motions. For instance, itis quite easy to see that a particle initially at rest with a non-zero acceleration will have a velocityincreasing exponentially in time with the time constant τ . These ‘runaway solutions’ are totally non-physical (all approximations fail in this case of course). We may use only this force when it is a smallperturbation of the motion.

Even then, it is not in a very convenient form. Assuming that the motion is nearly harmonic, weget easily a ≈ −ω2

0r and, hence,

fr = −mτω20

dr

dt. (1.8)

Injecting this approximate form in the equation of motion we get:

d2r

dt2+ γ

dr

dt+ ω2

0r = 0 , (1.9)

a standard damped harmonic oscillator motion, with:

γ = ω20τ. (1.10)

2Obtained by equating the electron mass energy mc2 with that of a uniform spherical charge distribution with radiusre and total charge q.

1.1. A CLASSICAL MODEL: THE HARMONICALLY BOUND ELECTRON 11

being the amplitude damping coefficient (the energy damping coefficient is obviously 2γ).Let us compute the order of magnitude of γ. We need to get a reasonable order of magnitude of

ω0. For that, let us us inject a bit of quantum mechanics in this classical model: ω0 ≈ Ry/h, where Ris the Rydberg constant. We then note that R = mc2α2/2, where

α =e2

4πε0hc≈ 1

137(1.11)

is the dimensionless fine structure constant, measuring the strength of the electromagnetic interactionin quantum units. We have then

γ

ω0= ω0τ =

Rτ

h=α3

3≈ 1.3 10−7 . (1.12)

We thus get the right order of magnitude. The quality factor of the electron motion is of the order of107 (note that our choice of ω0 corresponds to a deep-UV transition for which the lifetime is notablyshorter than for the ordinary optical transitions in alkali atoms for instance). Note here that γ is thedamping rate for the motion amplitude. The damping rate of the mechanical energy is twice as large.

1.1.2 Polarizability

The steady state for the free evolution is clearly an electron at rest, not a very interesting situa-tion. We thus study now how the atom responds, in the steady state, to an oscillating electric fieldE0uz exp(−iωt) (that of a plane wave or, more realistically, of a laser beam), where we assume withoutreal loss of generality that the incident field is polarized along the z axis and that the amplitude E0

is real. This field can be at a frequency different from that of the atomic oscillator. The equation ofmotion is now:

d2r

dt2+ γ

dr

dt+ ω2

0r =qE0

muze−iωt , (1.13)

In the steady state, r = r0 exp(−iωt). We can write the oscillating dipole created by the movingelectron as d = d0 exp(−iωt), with

d0 = qr0 = ε0αcE0uz , (1.14)

where

αc =q2

mε0

1

ω20 − ω2 − iγω

, (1.15)

is the classical polarizability of the atom at the frequency ω (note that the polarizability has thedimension of a volume). This polarizability has an evident resonant behavior around the atomicresonance frequency. We will now use this classical expression of the polarizability to discuss thediffusion of radiation by an atom, an extremely important problem practically.

1.1.3 Diffusion by an atom

The electron set in motion by the field of the incoming wave. In turn, it radiates in all directions (withthe emission diagram of a dipole). The total average radiated power P is, according to the Larmorformula, proportional to the average square acceleration ω4|r0|2/2. Hence

P =1

2mτω4|r0|2 , (1.16)

or, using the polarizability and the definition of τ

P =ε0

12πc3|αc|2ω4E2

0 (1.17)


If the incoming field is close to a plane wave, the incident power per unit surface is Pi = ε0cE20/2. We

thus define a surface as the ratio σc = P/Pi, called the diffusion cross-section:

σc =1

6π

(ω

c

)4

|αc|2 =8π

3r2e

ω4

(ω20 − ω2)2 + γ2ω2

, (1.18)

where we have introduced again the classical electron radius, re.Since γ is rather small, σc has a sharp resonance at the atomic frequency. We may thus distinguish

three important limiting cases, corresponding to three famous diffusion regimes:

• The Rayleigh diffusion, for ω < ω0 and ω0 − ω γ. We then get

σc =8π

3r2e

ω4

ω40

. (1.19)

The cross section is rather small, of the order of the surface of an electron, but it increasesrapidly with the frequency. This is the situation encountered in the diffusion of solar light bythe atmosphere. The optical resonances of the nitrogen and oxygen molecules are mainly in theUV and the visible light is thus in the Rayleigh regime. The shorter wavelengths are diffusedmuch more efficiently than the red, because of the ω4 dependence. This is why the sky is blue3

and why the solar light at sunset is red, since most of the blue has been diffused out. We caneasily estimate the attenuation length of visible light in the atmosphere. With σc ≈ 10−30 m2, anumeric density of N = 1025 m−3, the attenuation length is L = 1/Nσc ≈ 100 km, a reasonableestimate for the atmosphere thickness crossed by light at sunset.

• The Thomson diffusion corresponds instead to ω > ω0. We then get

σc =8π

3r2e . (1.20)

The value is small and the attenuation accordingly weak. Most materials are nearly transparentabove the resonance frequency.

• The most interesting regime is of course the resonant case when ω ≈ ω0. We can use this nearequality to simplify the expression of the cross section, and get finally

σc =8π

3r2e

ω20

4(ω0 − ω)2 + γ2. (1.21)

The cross section has a Lorentzian resonance, with a width γ. At exact resonance ω0 = ω, weget

8π

3r2e

ω20

γ2. (1.22)

Using the previous discussions, we note that

reω0

γ=

3

2cτ

1

ω0τ=

3

4πλ0 , (1.23)

where λ0 = 2πc/ω0 is the wavelength of the resonant radiation. Hence

σc =3

2πλ2

0 . (1.24)

3It is not that a simple phenomenon. The short scale fluctuations of the density of air at the normal pressure playan important role to understand why the sky is diffusing at all.

1.1. A CLASSICAL MODEL: THE HARMONICALLY BOUND ELECTRON 13

The cross section is close to the square of the resonant wavelength, much larger than the sizeof the electron, or even than that of the atom. An atomic vapour is thus expected to be quiteopaque close to resonance, since the distance between atoms is typically much less than λ0. Thepresent model has however a limited validity at resonance; Being purely linear, is assumes thatthe atom can radiate as much power as we send inside the cross section area. This is far frombeing true, as we will see in the next Chapter. Nevertheless, this classical order of magnitudeestimate is correct as long as the radiated power remains small compared to what we will callthe ‘saturation power’, corresponding roughly to the diffusion of photons at a rate γ. The orderof magnitude of the saturation intensity is a milli-Watt per square centimetre.

1.1.4 Propagation in matter

Before switching to the Einstein’s coefficients, we will use the atomic classical polarizability to discussbriefly the propagation of a field in matter. This will provide us with an insight in the loss/gainphenomenon that will be useful for the next Section. Let us first recall the basics of Maxwell equationsin matter. We neglect here the magnetic properties of the matter (magnetic effects are generally quitesmall unless the matter is a ferromagnet, generally not very transparent). The electric displacementD is

D = ε0E + P , (1.25)

where P is the density of polarization of the bound charges. For a collection of independent moleculesat a low enough numerical density N (so that the field produced on each molecule by its neighboursis much smaller than the incident field), we have P = ε0χcE, where χc = Nαc is the dimensionlesssusceptibility. Hence, D = ε0εrE where εr = 1 + Nαc is the relative permittivity (note that ourexpressions are only valid in the Nαc 1 regime). Injecting this relation in the Maxwell equationsand playing the standard tricks, we get the modified d’Alembert equation ruling the propagation ofthe amplitude of a wave oscillating at frequency ω:

∆E +ω2

c2εrE = 0 . (1.26)

Subsituting a plane wave E = E0 exp(ik · r) in this equation, we finally obtain the dispersion relation:

k2 = k20εr , (1.27)

where k0 = ω/c is the free-space propagation wavevector. We introduce the medium’s refraction indexn as n =

√εr and get k = nk0. Noting the real and imaginary parts of n as n = n′ + in′′ (we will use

these notations for all complex quantities), we get:

n′ =1√2

√ε′r +

√ε′2r + ε′′2r and n′′ =

ε′′r√2

1√ε′r +

√ε′2r + ε′′2r

. (1.28)

The real part of the wavevector and of the index correspond to the standard dispersive phaseaccumulation upon propagation. The imaginary part describes the exponential attenuation (or am-plification) of the amplitude of the electric field. We immediately read on the previous equations thatthe attenuation is directly linked to the imaginary part of the polarizability (when it is real, the indexis real and the amplitude constant). We can give a simple interpretation of this observation in termsof matter-field energy exchange.

In the propagation, the density of energy (energy per unit volume) given by the field to the boundcharges is the scalar product of the field by the bound current density, itself the time derivative ofthe polarization. Noting E0 and P0 the local amplitudes of the electric field and polarization waves,


we can write the average density of energy released to the matter, 12Re j0 ·E0∗, where the current j is

j = ∂P/∂t and its amplitude j0 = −iωP0:

E =1

2Re (−iωP0 ·E∗0) , (1.29)

where the star indicates a complex conjugation. The real part of the susceptibility disappears fromthis expression, as expected, and

E =1

2ε0ωχ

′′|E0|2 =1

2ε0ωNα

′′|E0|2 . (1.30)

As expected, the energy released by the field into the matter is proportional to the imaginary part ofthe polarizability. For the elastically bound electron model,

α′′ =q2

mε0

γω

(ω20 − ω2)2 + γ2ω2

(1.31)

is always positive. The energy flows from the field into the matter. The field is absorbed (and infact diffused in other directions since γ only describes here the radiation damping). Accordingly, itpropagates as an evanescent field in the medium. There is no possible amplification of a light field inthe steady state in our classical mode. The Laser effect is thus, as we will see in the next Chapter, apurely quantum one.

1.2 Einstein’s coefficients

We give in this Section a short account of the phenomenological coefficients proposed by Einsteinto describe the matter-field energy exchanges. They predict the stimulated emission process, whichpermits, in special conditions, the amplification of a light field in matter and, hence, the laser effect.These coefficients are also important since we will find them again as simple limiting cases of thesemi-classical model in the next Chapter.

1.2.1 The three coefficients

We plan to describe the energy exchanges between matter and field. The field is contained in a box.We will forget all about the field coherence, phase information, interferences, and assume that is hasa wide spectrum. In fact, Einstein used this model to describe the matter-field thermal equilibrium.The blackbody radiation has a wide spectrum and its coherence properties are clearly of no relevance.It will be entirely described by its spectral energy density uv. The energy per unit volume betweenfrequencies4 ν and ν + dν is uνdν, and uν is measured in J/m3Hz. We will assume, for the sake ofsimplicity, that this energy density is uniform across the volume of the box. The total electromagneticenergy per unit volume is u =

∫uv dν. In qualitative quantum terms, the numerical density of photons

between ν and ν + dν is uν/hν.The matter is made up of atoms with two levels, e above g, having respective energies Ee and Eg.

We assume them first to be non-degenerate, for the sake of simplicity. These levels are coupled to thefield: the atom may undergo a quantum jump from e to g and release a photon (emission process) orcan absorb a photon and make the transition from g to e. The resonant frequency of this transitionwill be denoted as (Ee − Eg)/h = ν0. The energy difference between the two levels is thus hν0. Thiswill be the only quantum part in our model. The constant total number (or density) of atoms is N ,the numbers of in the upper state Ne (Ng for the lower state) are normalized to N so that Ne+Ng = 1.Note that we could use as well the total number of atoms, proportional to the density since we madean homogeneity hypothesis.

4For historical reasons we will use here the frequencies and not the angular frequencies.

1.2. EINSTEIN’S COEFFICIENTS 15

Our goal is to write a rate equation for the time evolution of the densities Ne and Ng. We havefor that to consider three processes:

• Spontaneous emission. The atoms in the upper state e can spontaneously decay to the lowerstate g by emitting a photon, as in the classical model of the last section. This process occurswith a constant probability per unit time Aeg, that we will also denote by Γ for consistency withthe next Chapters. The rate equation for the evolution of Ne under this process is thus:

dNe

dt

)spon

= −AegNe . (1.32)

• Absorption of radiation. An atom can certainly absorb radiation and make a transition fromthe lower to the upper state. We will use here a cross section argument and assume that theprobability for this transition to occur is proportional to the total power at the atomic frequency,i.e. to the spectral density of energy at the atomic frequency. Noting Bge the proportionalitycoefficient, we can write the rate equation of Ne due to this sole process as:

dNe

dt

)abs

= Bgeuν0Ng . (1.33)

Einstein noticed that these two processes alone (absorption and spontaneous emission) do nottell the whole story. Imagine that the radiation is near thermal equilibrium at a very hightemperature T . Whatever the atomic frequency ν0, uν0 becomes very large. The steady stateof the two rate equations written so far is Ng = 0. All atoms are in the upper state! This isclearly not the thermal equilibrium: at a very large temperature, we expect to find only half ofthe atoms in e. The matter does not get in thermal equilibrium with the field. This cannot beaccepted. We must thus add a third process to our list.

• Stimulated emission. Einstein, for the first time, supposed that a third process may causeatomic transitions. The atom can release a photon, and jump from e to g under the influence ofthe already present radiation, with a rate also proportional to the energy density at the atomicfrequency. This is described by an additional term in the equation for Ne:

dNe

dt

)stim

= −Beguν0Ne . (1.34)

The rate equation for the excited state atomic density thus writes finally:

dNe

dt= −AegNe −Beguν0Ne +Bgeuν0Ng . (1.35)

In this equation, only the spontaneous emission rate can be computed easily with the classical modelof the radiation reaction force. We will see, however, that the condition of the matter-field thermalequilibrium results in relations between these coefficients.

1.2.2 Relations between the coefficients

Let us thus assume that the radiation and the matter (the atoms) have reached a thermal equilibriumat temperature T . The densities of atoms in e and g must thus obey the Boltzmann statistics:

Ne

Ng= e(Eg−Ee)/kbT = e−hν0/kbT , (1.36)

where kb is the Boltzmann constant.


The spectral density of radiation is given at the same time by the Planck law5:

uν0 =8πhν0

3

c3

1

exp(hν0/kbT )− 1. (1.37)

In the steady state, these two equilibrium conditions plugged in Eq.(1.35) should lead to dNe/dt =0. In order to make the algebra simpler, let us first consider the T → ∞ limit. Then, uν0 → ∞ andNe/Ng → 1. The spontaneous transitions are negligible in this high radiation power limit, and weimmediately get that the absorption and stimulated emission coefficients are equal:

Bge = Beg = B (1.38)

Noting for simplicity Aeg = A, we get the steady state at a finite temperature T when:

A+Buν0 = Buν0ehν0/kbT (1.39)

and hence

uν0 =A

B

1

exp(hν0/kbT )− 1(1.40)

A mere comparison with Planck law immediately leads to

A

B=

8πhν03

c3=

8πh

λ30

, (1.41)

where λ0 = c/ν0 is the resonant wavelength of the transition. All the Einstein’s coefficients can bededuced simply from the spontaneous emission one, a quite remarkable result.

Let us now briefly evoke the case of degenerate atomic levels. It happens quite often in atomicstructure that a few quantum levels have the same energy. This is for instance the case for allmagnetic sublevels of an atomic state, when no external static electric or magnetic field is applied.Let us denote by ge and gg the degeneracies (dimension of the energy eigenspaces) of energies Eeand Eg. The Boltzmann thermal equilibrium condition now reads Ne/Ng = (ge/gg) exp(−hν0/kbT ).Getting to the infinite temperature limit, we find Bge/Beg = ge/gg. Assuming then (sometimes a grossapproximation) that we can define a single spontaneous emission rate from e to g, we can compute theB coefficients from Aeg. We leave this to the reader as an exercise. We will not consider this purelyalgebraic complication any further and assume that all levels are non-degenerate in the following.

1.2.3 Stimulated emission: the laser

The novelty of Einstein’s approach is the introduction of the stimulated emission. This leads naturallyto the possibility of amplifying radiation by the propagation in a medium, an impossible feat in theharmonically bound charge model. Let us consider thus a plane wave at the resonant frequency ν0

impinging on a thin slice of atoms, with the densities Ne and Ng. Let P be the incident power per unitsurface. At the exit of the medium, the power is P + dP. The increment dP reflects the matter-fieldenergy exchange. Let us assume (a good approximation for most lasers) that spontaneous emissionis negligible (that uν is large enough so that the absorption and stimulated emission processes rulethe evolution). The power increment δP is thus proportional to the difference between the stimulatedemission and absorption rates. Since uν is obviously proportional6 to P, we get finally

dP ∝ P(Ne −Ng) = P D , (1.42)

5It will be derived later in Chapter 3. We take it here as granted.6In this simple model, we will give explicitly the proportionality coefficients.


where we introduce the population inversion density D = Ne − Ng. For a constant D, we thus findfor the propagation through a macroscopic thickness of material either an exponential decrease of thepower (evanescent wave) or an exponential increase (gain!).

A medium with a gain, an amplifier for light, naturally leads to the idea of a light oscillator. Inthe electronic realm, an amplifier can be turned into an oscillator by connecting the input to the(non-inverting) input. The same applies for Light Amplification by Stimulated Emission of Radiation(LASER). A gain medium placed between two highly reflecting mirrors (an optical cavity) will sustaina steady state oscillation as soon at the single-pass gain through the medium exceeds the round-triplosses of the cavity. Provided this threshold condition is met, the laser can produce an intense field inthe steady state.

The condition to obtain a gain is that the population inversion D should be positive, i.e. thatthe upper level should be more populated than the lower one. It is quite clear that this populationinversion condition is never met at thermal equilibrium7. The laser can thus operate only if we providea fast pumping mechanism of the upper level, together with a fast drain of the population in the lowerone.

There is no way to achieve that in the steady-state with a two-level structure. The most commonsituation is to use a four-level structure. The lower level g is then above the real ground state fand connected to it by a fast (radiative or non-radiative) damping mechanism. From the real groundstate f , a pumping process (excitation by light, electronic collisions...) populates an excited statei, lying above e. A fast relaxation empties i into e. When the relaxation processes are faster thanthe laser-field induced transition, and when the pumping from f to i is fast enough, the level e mayreach a steady-state population above that of g, achieving the condition for laser amplification. Otherschemes are possible, with a three-level structure (as in the He-Ne laser), with unstable ground states(excimer lasers), or with excitations made up of annihilating particles (semiconductor lasers)

1.2.4 A simple model of laser radiation

We give here a simple model of the operation of a four-level laser. We will omit all the hairy propor-tionality factors and concentrate on the physical contents. The variables of interest are the populationinversion D and the intra-cavity intensity I, measured for instance in terms of the volumic density ofphotons in the cavity, and obviously proportional to uν . In a four-level configuration, the level g isstrongly damped and practically empty at all times. We thus have to a good approximation D ' Ne.The rate equation for the intensity is of the form:

dI

dt= −κI + gID , (1.43)

where the term κI models the cavity losses (for instance through one of the cavity mirrors used as anoutput coupler). The gain term is proportional to I (through uν0) and to the population inversion D.The gain coefficient g encapsulates the geometry of the medium, the Einstein coefficients a.s.o. In thesame vein, we can put the equation of motion of D under the form:

dD

dt= Λ− ΓD − gID . (1.44)

In this equation, the first term Λ describes the pumping mechanism that populates the upper level e.The last term expresses that any gain in intensity is made at the expense of the population inversion.Finally, the mid term −ΓD reflects the spontaneous damping of the population inversion under theaction of the e state relaxation (including spontaneous emission outside of the cavity).

We thus have two coupled equations of motion and we look for a steady state. The equation for Ialways admits I = 0 as a solution. The ‘laser off’ condition is always possible. If I = 0, we get from

7The level degeneracies, when taken into account, do not change this fundamental conclusion.


Eq.(1.44) D = Λ/Γ. The population inversion rises linearly with the pumping rate Λ, an intuitiveresult

This operating point is fortunately not the only one. The equation of evolution of I admits anon-zero solution if D = κ/g. Plugged in the population inversion equation, this leads to:

I =1

κ

(Λ− Γκ

g

). (1.45)

This solution is physically relevant only if I ≥ 0 i.e. if

Λ ≥ Λt =Γκ

g. (1.46)

For Λ < Λt, the only solution is thus I = 0. Above the ‘laser threshold’ i.e. for Λ > Λt, two solutionscoexist, I = 0 and I 6= 0. A simple stability analysis shows that the I = 0 solution is unstable abovethreshold. Any small intensity fluctuations (for instance those due to the few spontaneous emissionphotons trapped in the cavity mode) will trigger an evolution towards the high intensity solution. Itis clear that the threshold condition expresses that the laser gain exceeds the losses (due to the Γ andκ coefficients). Above the threshold, the laser intensity grows proportionally to the pumping rate.The population inversion remains constant, ‘clamped’ at the value Dt = κ/g. Note that our model isoversimplified. A practical laser unfortunately cannot give an infinite power when pumped infinitelystrongly!

1.2.5 Properties of laser radiation

In this Einsteinian approach, we have only thought in terms of energies, in terms of number of photons.We have not taken into account the coherence properties of the light field, i.e. the phase information.This is not a lecture on lasers and we will refer the reader to the many textbooks on lasers for amore complete description. Let us just get a qualitative insight. The stimulated emission process,interpreted in terms of photons, corresponds to an atom in e emitting a photon under the influenceof an incoming, ‘triggering’ photon. The two photons must belong to the same field mode. We willdefine the notion of field mode properly in Chapter 3. Let us just say that if the incoming photon isin a plane wave, the stimulated photon is also in the same plane wave. It has, whatever it means, thesame frequency, phase, direction of propagation and polarization as the incoming photon. This is, bythe way, a simple example of ‘bosonic amplifications’. Photons are bosons and love to share the samequantum state.

The laser radiation is thus (the ‘thus’ is somewhat a short-cut through the complete theory)coherent. It is spatially coherent: different points in the laser beam have fixed phase relationships.The beam can thus be extremely directive, be focused in tiny spots with a size of the order of thewavelength, much smaller than those obtained with incoherent sources. The beam is also coherent intime. The optical phase can be stable over very long times, making it possible, for instance, to realizeinterferometers with extraordinarily long optical path differences (tens of thousand of kilometres forthe gravitational wave antennas).

The lasers are powerful, up to 100s of kW for the welding and cutting lasers. Ultra-short pulsescan be generated, in the fs range, with peak powers reaching in the PW range (1015). WIth suchpowers, the atoms are immediately ionized and the electrons reach relativistic velocities in one opticalperiod. This leads to the efficient generation of very high harmonics, due to the non-linearity of therelativistic motion.

Lasers can be ultra-stable, with linewidths now in the mHz range. This is of course instrumentalfor the realization of clocks based on the measurement of a very narrow optical transition. The bestclock in 2014 is based on cold atoms in an optical lattice, interrogated by ultrastable lasers. It providesa frequency stability of 10−17 (a couple of seconds over the age of the Universe!).


Finally, semi-conductor diode laser are everywhere in our lifes, in the CD or DVD players but alsoin the optical fibre communication systems. Without them, there would be no internet!

Chapitre 2

Semi-classical approach

The first part of the previous Chapter proposed a classical model of atom-field interaction. It is ofcourse quite limited and, for instance, does not predict or support the possibility of light amplification.We will go one step beyond in this Chapter by studying a semi-classical model for the matter-fieldcoupling. We will use a classical radiation model, entirely described by the electric and magnetic timeand space-dependent fields, ruled by Maxwell’s equations. But we will treat the atom as a quantumobject, whose energy levels are solution of the Schrodinger equation. We will obtain with this modelmany important results, which are valid provided the quantum nature of the field can be neglected.In a nutshell, they are valid as long as the number of photons in the field is very large and as long asthe field quantum state is close to a classical one (a coherent state in technical terms, much more onthat later).

The first point in this Chapter will be to get the atom-field interaction Hamiltonian. We will pro-pose two equivalent forms, equally useful in the following. We will then briefly describe the interactionof an atom with a weak non-resonant field (whose frequency is different from that of all the atomictransitions). We will see that we recover in this way most of the results of the harmonically boundmodel, renewing the interest of this simple approach.

We will then turn to the more important problem of an atom interacting with a resonant field,when the atom can be safely assimilated to a two-level system. We will explore first the case of afree atom, without relaxation, and investigate the Rabi oscillation mechanism and its applications tohigh-precision spectroscopy and atomic interferometry (Ramsey method). We will then learn how todescribe atomic relaxation, either due to spontaneous emission or to other perturbing effects.

The last part of the Chapter will then be devoted to an in-depth analysis of the interaction of anrelaxing atom with a resonant field. We will derive the Optical Bloch equations (OBE) describing theevolution of the atomic density operator and review briefly a few applications.

2.1 Interaction Hamiltonian

We consider here, for the sake on simplicity, a one-electron atom (the hydrogen atom). The resultswe will get, for most of them, do not depend upon this assumption. They hold, within some algebraicvariations, for many-electron atoms. They also hold perfectly for alkali atoms, whose outer electronis the only one taking part in optical transitions. The Hamiltonian of the free atom is

H0 =P 2

2m+ qU(R) , (2.1)

where P and R are the momentum and position operators of the electron (mass m, charge q)1.The potential U is the electrostatic Coulomb potential binding the electron to the nucleus or to the

1As much as possible, we will use upper case letters for the operators in the Hilbert space, and lower-case letters for thepositions and momentum eigenvalues. With this convention, the definition of the position eigenstates reads R |r〉 = r |r〉.

21

22 CHAPITRE 2. SEMI-CLASSICAL APPROACH

core (nucleus and other electrons). The eigenstates |i〉 of H0 are defined by the eigenvalue equationH0 |i〉 = Ei |i〉. The ground state will be called |g〉.

We place this atom in a radiation field defined by its potential vector A(r, t), its electric potentialV(r, t), its electric and magnetic fields E and B. Obtaining the complete atom-field Hamiltonianis as simple as replacing all positions and momenta by operators in the expression of the classicalHamiltonian of a charge in an electromagnetic field, and hence:

H =1

2m(P− qA(R, t))2 + qU(R) + qV (R) , (2.2)

where A(R, t) is the electron Hilbert space operator obtained by merely replacing r by R in theclassical vector potential (there is no commutation relation problem involved in this operation). Notethat we neglect, in all these expressions, the electron spin and its interaction with the incomingmagnetic field. The effects of spins and magnetic fields are generally quite negligible when there is aresonance between an atomic transition and the incoming electric field.

2.1.1 Gauge choice

The Hamiltonian given by Eq.(2.2) is rather impractical. We will start simplifying it by making aproper gauge choice for the electromagnetic field.

Let us recall briefly that the fields only are physical quantities. The potentials can be changed ina gauge transformation:

A′ → A = A′ + ∇χ(r, t)

V ′ → V = V ′ − ∂χ

∂t, (2.3)

where χ is an arbitrary function of space and time, without altering the physical fields. We can thusimpose for our benefit an extra condition to the vector potential. In electromagnetism, it is customaryto use the Lorentz gauge condition ∇ ·A + (1/c2)∂V/∂t, since it is manifestly invariant in a Lorentztransformation. It is an easy exercise in vector calculus to show that whatever A′, V ′, χ can be chosento satisfy this condition.

We are dealing here with non-relativistic quantum mechanics. We do not care about Lorentzinvariance and we are thus free to use the much simpler Coulomb gauge:

∇ ·A = 0 , (2.4)

(note that χ is obtained in this case by solving a Laplacian equation).In order to explore the benefits of this gauge choice, we will introduce the time-space Fourier

transform of the vector potential, A(k, ω), defined by:

A(r, t) =1

4π2

∫A(k, ω)ei(k·r−ωt) dkdω . (2.5)

The vector field A(k, ω) can always be decomposed in a component parallel to k, A‖, and one perpen-dicular to it, A⊥. The potential vector can thus itself be decomposed in a ‘longitudinal’ component,A‖(r, t), Fourier transform of A‖, and a ‘transverse component, A⊥(r, t), Fourier transform of A⊥2.

With this definition, the space-time Fourier transform of ∇ · A is obviously ik · A. Since it isidentically zero, we deduce that the potential vector is ‘transverse’, its Fourier transform reducing tothe A⊥ component and that the vector potential is purely transverse: A = A⊥. The electric field isa transverse vector also (in terms of its space-time Fourier transform E), since its divergence is zero

2Note that the ‘longitudinal’ or ‘transverse’ nature applies to the Fourier transforms, but has no direct simple geo-metrical consequence for the transverse and longitudinal potential vectors themselves.

2.1. INTERACTION HAMILTONIAN 23

in a place free of charge (the incoming wave is external to the electron, and the electron does notparticipate to it as a source).

Using the expression of the electric field:

E = −∂A

∂t−∇V , (2.6)

and noting that the gradient of V has a Fourier transform parallel to k (longitudinal), we get obviously∇V = 0 in the Coulomb gauge. Since a constant potential does not imply any physical effect, we canfinally set the potential V to zero: the electrostatic potential of an electromagnetic wave is identicallyzero in the Coulomb gauge.

2.1.2 A ·P interaction

We now proceed to expand the (P− qA(R, t))2 quadratic term in the full Hamiltonian (2.2). We haveto take care that P does not commute with A, which is a function of the position operator R. Herealso, the Coulomb gauge will help us.

It is easy to show indeed that

[Pi, f(R)] = −ih ∂f∂Ri

, (2.7)

where i stands for x, y or z. Hence,∑i

[Pi, Ai] = −ih∑i

∂Ai∂Ri

= −ih∇ ·A = 0 . (2.8)

andP ·A =

∑i

PiAi =∑i

AiPi = A ·P . (2.9)

The complete Hamiltonian can thus be written exactly as:

H =P 2

2m+ qU(R)− q

mP ·A +

q2

2mA ·A . (2.10)

The remaining quadratic term is quite an annoyance and makes the problem extremely difficult.Fortunately, it is generally quite small compared to the term linear in A if the incoming wave electricfield is small as compared to the static field binding the electron on its orbit (one atomic unit ofelectric field, about 1011 V/m). This is nearly always the case, except with the very high intensitylasers that we evoked in the end of the previous Chapter. We will thus, for all practical interactionsof an atom with a reasonable laser, neglect the quadratic terms and write the Hamiltonian as:

H = H0 −q

mP ·A(R, t) . (2.11)

The coupling term remains quite complex, since it involves both the momentum and the positionoperators. We can make a further simplification by performing the ‘dipole approximation’. It relies onthe fact that the resonant wavelength of an optical atomic transition, of the order of the micrometer,is much larger that the size of the atom itself, a fraction of a nanometre. Hence, we can assimilate thepotential vector at the position of the electron to that at the centre of the atom, that we take fromnow on as the origin. We then get the final Hamiltonian in the in the ‘A ·P’ form:

H = H0 −q

mP ·A(0, t) , (2.12)

where the coupling term is now simply proportional to the momentum operator. Note that the dipoleapproximation might not be valid if we study the interaction of non-resonant light with an atom.Diffusion of a X-ray radiation by an atom cannot be treated in this way, since the wavelength is ofthe order of the size of the atom itself.


2.1.3 D · E interaction

In the previous paragraph, we have obtained the coupling Hamiltonian in a simple form. However,it is not the most intuitive one. We are used, since the previous Chapter, to think of the atom asan electric dipole. A dipole interacts with the field by a −d · E energy. Moreover, this form of theinteraction Hamiltonian would be manifestly gauge-invariant since it only involves the ‘physical’ fieldE and not the vector potential as before. Is there a simple way to cast the full quantum Hamiltonianunder this form? In fact, there are two ways that we will expose now since they shed complementarylight onto the problem.

The Goppert-Mayer transformation

We restart from the full Hamiltonian of Eq.(2.2) before switching to the Coulomb gauge, includingthus the scalar potential of the incoming wave and we perform the dipole approximation on thisHamiltonian before all other operations. We thus replace the vector potential by its value at theorigin, A(0, t)3. We also neglect the quadratic terms.

We will be a bit more careful with the static potential, and expand it to the first order in R asV = V (0, t) + R ·∇V (0, t). We have to include this first order terms, since the potential vector actsto order zero in R, while a uniform electric potential (at order zero) has no effect. We can forget theuniform term in the potential, and write the electric dipole approximation Hamiltonian as:

H = H0 −q

mP ·A(0, t) + D ·∇V , (2.13)

where we introduce the dipole operator D = qR.

By choosing at this point the Coulomb gauge, we could cancel the electric potential and its gradientand recover the A ·P interaction. We make instead, following Maria Goppert-Mayer, a different choiceof gauge. We replace A by A′ and V by V ′, using the gauge transformation expression Eq.(2.3) (andexchanging the roles of A and A′). We chose χ(r, t) so that the new vector potential A′(0) cancelsat the origin. The gauge function χ(r, t) = −r · A(0, t) does the job since its gradient is precisely−A(0, t). The new scalar potential is then given by:

V ′ = V + r · ∂A(0, t)

∂t, (2.14)

Its gradient at the origin is then simply:

∇V ′(0) = ∇V (0) +∂A(0, t)

∂t= −E(0) , (2.15)

the actual electric field at the origin. In this new gauge the Hamiltonian just writes:

H = H0 −D ·E(0) , (2.16)

the required form for the interaction of a dipole with a field. This is the form that we will mostly usein the following (it is often more intuitive to work in terms of the position operator than in terms ofthe momentum one). Note that this expression is not valid outside the dipole approximation that wemade at an early stage of the calculation. In the frame of this approximation, it is, once again, clearlygauge invariant.

3Note that the vector potential at the origin is a scalar in the atomic Hilbert space and thus commutes with theelectron momentum.

2.2. NON-RESONANT INTERACTION: PERTURBATIVE APPROACH 25

Unitary transform approach

The other path to the D·E interaction does not use a gauge transformation, but a unitary basis changein the atom’s Hilbert space. We start from the general Hamiltonian Eq.(2.2), switch to the Coulombgauge (this cancels the incoming wave scalar potential), and perform the dipole approximation byreplacing A(r, t) by A(0, t). We get finally the Hamiltonian:

H =1

2m(P− qA(0, t))2 + qU(R) . (2.17)

We then perform a time-dependent basis change in the Hilbert space, defined by the unitary

operator T (t) (T †T = 11) and changing |Ψ〉 into∣∣∣Ψ⟩ = T |Ψ〉. It is easy to show, by a mere substitution

in the original Schrodinger equation, that the new wavevector evolves under the Hamiltonian:

H = THT † + ihdT

dtT † (2.18)

We now choose T (t) so that:

TPT † = P + qA(0, t) , (2.19)

which defines T as a time-dependent translation of the electron’s momentum. The unitary T is thusan exponential function of the position operator:

T = e−ihqR·A(0,T ) = e−

ihD·A(0,T ) , (2.20)

where we get back the dipole operator. We note that

T (P− qA(0, t))2 T † = P2 , (2.21)

and that

ihdT

dtT † = D · dA(0, T )

dt= −D ·E(0, t) , (2.22)

since the scalar potential vanishes in the Coulomb gauge. We also observe that T does not changeU(R) since T and R commute. We finally obtain:

H = H0 −D ·E(0) , (2.23)

where H0 is the field-free Hamiltonian. We get once again the interaction in the gauge-invariantelectric dipole form. We did not use here the weak field approximation (we did not neglect explicitlythe A2 term in the expansion of the Hamiltonian) and our expression is exact in the frame of thedipole approximation. It should be noted however that the unitary basis change also entails a changeof all physical observables in order to predict the same measurable quantities (O → TOT †) and thatthe non-linear terms in the potential are a priori present in these transformations. In the case of weakfields, these corrections do not contribute and we are left with an Hamiltonian fully equivalent to thatof the previous paragraph.

2.2 Non-resonant interaction: Perturbative approach

Before turning to the resonant atom-field interaction, we will study in this section the interaction ofthe atom with a non-resonant field in a perturbative regime. We thus assume that the atom is initially(t = 0) in its ground state |g〉. It interacts with a weak harmonic field, whose angular frequency ωis very different from that of all transitions to excited states connected to |g〉. The population of theexcited states will thus remain at all times much lower than that in |g〉. Without loss of generality,


we can assume that the incoming wave is polarized along the uz axis (we can use the superpositionprinciple to treat other polarization states) and write:

E(0, t) = E0uz cosωt . (2.24)

Using the electric dipole form for the interaction Hamiltonian, we get H = H0 + H1 with H1 =−qZE0 cosωt (Z being the uz component of the position operator). We now switch to an interactionrepresentation with respect to H0, defined by the free evolution operator U0 = exp(−iH0t/h). The

Hamiltonian in this representation is H = U †0H1U0. Let us decompose the interaction representation

state vector,∣∣∣Ψ⟩, on the basis |j〉 of the H0 eigenstates:∣∣∣Ψ⟩ =

∑j

βj |j〉 . (2.25)

Injecting this expansion in the Schrodinger equation and taking the scalar product of both sideswith a bra 〈k| (also an eigenstate of H0), we get:

ihdβkdt

=∑j

〈k|U †0H1U0 |j〉βj . (2.26)

Noting that U0 |j〉 = exp(−iωjt) |j〉, where ωj = Ej/h is the Bohr frequency of level |j〉, and settingωkj = ωk − ωj , we get:

dβkdt

= −qE0

ih

∑j

eiωkjt 〈k|Z |j〉βj cosωt . (2.27)

We have thus transformed our problem in a mere set of coupled first-order differential equations.However, this is a rather impractical infinite set (and even a continuous set is we include the continuumeigenstates of H0).

We now proceed to a first order approximation, a priori valid within our hypotheses. We assumethat all the amplitudes are small, except βg, which is close to one (and can always be assumed to bereal by a proper phase choice). Thus, only one term remains in the r.h.s of the previous equation and:

dβkdt≈ −qE0

iheiωkgt 〈k|Z |g〉 cosωt , (2.28)

a simple differential equation whose evident solution is:

βk(t) =qE0

2h〈k|Z |g〉

[ei(ωkg+ω)t − 1

ωkg + ω+ei(ωkg−ω)t − 1

ωkg − ω

]. (2.29)

We get the complete solution to the problem, with clear resonances at ω = ±ωkg i.e. at all the Bohrfrequencies connecting |g〉 to another level |k〉, provided that the matrix element 〈k|Z |g〉 does notvanish (selection rules).

It is instructive to compare this result with that of the classical model derived in Section 1.1. Wehad then computed the steady state dipole amplitude, d, for the harmonically bound electron anddefined the classical polarizability αc by d = ε0αc(ω)E. In order to compare αc with our quantumcalculation, we will estimate the average value of the dipole operator D = qZuz = Duz (by symmetrythe dipole operator has only a component along uz) in the state:

|Ψ〉 =∑k

βke−iωkt |k〉 . (2.30)

Note that we should quit the interaction representation for this calculation in order to get the timeevolution right. We get:

〈D〉 =∑`,k

β∗`βke−iωk`t 〈`| qZ |k〉 . (2.31)

2.2. NON-RESONANT INTERACTION: PERTURBATIVE APPROACH 27

Keepıng only the first oder terms, including at least one βg, this reduces to:

〈D〉 =∑k

βke−iωkgt 〈g| qZ |k〉+ c.c. . (2.32)

Injecting finally the explicit expression of βk and after a bit of algebra, we have:

〈D〉 =q2E0

2h

∑k

| 〈g|Z |k〉 |2[eiωt − e−iωkgt

ωkg + ω+e−iωt − e−iωkgt

ωkg − ω+ c.c.

](2.33)

We get a quite complex expression, with terms evolving at the drive frequency ω and other oscillatingat all the atomic Bohr frequencies ωkg. The latter are clearly an artefact in our calculation. We havenot introduced any damping and, hence, the excited states of our atom are everlasting. Even if we donot yet know how to treat it (much more on that later), there is a damping and the Bohr frequencyterms must vanish, leaving us with an evolution only at the drive frequency, as in the classical model.This evolution is simply given by:

〈D〉 =q2E0

2h

∑k

| 〈g|Z |k〉 |2[

eiωt

ωkg + ω+

e−iωt

ωkg − ω+ c.c.

]. (2.34)

We define the real quantum polarizability at frequency ω, αQ(ω), by 〈D〉 = ε0αQ(ω)E0 cosωt andget:

αQ(ω) =2q2

hε0

∑k

| 〈g|Z |k〉 |2 ωkgω2kg − ω2

. (2.35)

Let us recall here the classical polarizability in the case of a non-resonant frequency and negligibledamping γ (and hence in a situation where the polarizability is real):

αc(ω, ω0) =q2

mε0

1

ω20 − ω2

, (2.36)

where ω0 is the classical resonance frequency.

The two expressions are very similar. In fact, the quantum polarizability appears as a weightedsum of classical polarizabilities for resonant frequencies equal to the Bohr frequencies:

αQ(ω) =∑k

fkgαc(ω, ωkg) , (2.37)

where we introduce the real, positive, dimensionless ‘oscillator strength’:

fkg =2mωkgh| 〈g|Z |k〉 |2 . (2.38)

In this picture, an atomic medium of numeric density N appears a a mixture of classical harmon-ically bound electrons with resonance frequencies ωkg and densities Nfkg. All our conclusions on thepropagation of light in the classical medium thus retain their validity in this perturbative semi-classicalmodel. This picture holds, in particular, because the oscillator strengths obey the important sum rule:∑

k

fkg = 1 , (2.39)

that we now proceed to establish. Let us rewrite fkg as

fkg =2mωkgh〈g|Z |k〉〈k|Z |g〉 . (2.40)


Noting that

[Z,H0] =ih

mPz , (2.41)

(since only the momentum term in H0 contributes to the commutator), we have

〈k|Pz |g〉 =m

ih〈k|ZH0 −H0Z |g〉 = −mωkg

i〈k|Z |g〉 . (2.42)

Using this relation to replace the matrix element 〈k|Z |g〉 in the oscillator strength, we get

fkg =2

ih〈g|Z |k〉〈k|Pz |g〉 . (2.43)

The sum over k introduces a closure relation and:∑k

fkg =2

ih〈g|ZPz |g〉 . (2.44)

The oscillator strength being real, the r.h.s of this equation is also real and equal to half the sum ofit with its complex conjugate. Hence∑

k

fkg =1

ih〈g|ZPz − PzZ |g〉 = 1 , (2.45)

completing the sum rule demonstration.

2.3 Free atom and resonant field

The perturbative approach of the last Section only holds when the incoming radiation has a lowintensity and is non resonant (the results boldly diverge when the incoming field hits an atomicresonance). We will now consider the experimentally very important case of a monochromatic fieldwhose frequency ω is extremely close to that, ω0 = ωeg of one atomic transition, between the two levels|g〉 (lower, possibly ground level) and |e〉. We will assume that these two levels are non degenerate toextract the physics without the algebraic complications of a multi-level structure.

In this situation, assuming furthermore that the atom is initially either in |g〉 or in |e〉, it is clearthat all the other atomic levels play a minute role, the main effect of the incoming wave being toinduce transitions between the two levels. We will first consider in this paragraph an ideal two-levelatom, undergoing no relaxation mechanism and no perturbation from the outside world, besides theincoming wave of course.

2.3.1 Atomic system and interaction

Spin-1/2

Let us first briefly remind the notations for the quantum description of this simplified two-level atom.The Hilbert space is of dimension 2, and the system isomorphic to a spin-1/2 system placed in aconstant magnetic field. We will thus assimilate |e〉 and |g〉 with the eigenstates |+〉 and |−〉 of thePauli σz operator. A basis of the real vector space of Hermitian operators for the spin is the set ofPauli operators which write in matrix form in the |+〉 , |−〉 basis:

σx =

(0 11 0

); σy =

(0 −ii 0

); σz =

(1 00 −1

); 11 . (2.46)

Note that, in quantum information science, a quite active field worldwide, a spin-1/2 is the elementaryinformation carrier, a qubit. In this context, the |+〉 and |−〉 states are the ‘logical’ states of the qubit

2.3. FREE ATOM AND RESONANT FIELD 29

and are often referred to as |0〉 and |1〉. We will use from time to time these quantum informationnotations when useful.

All the Pauli matrices are hermitian, with square equal to 11 and are thus also unitary. They obeythe standard commutation relation of angular momentum operators:

[σx, σy] = 2iσz , (2.47)

and all the similar commutations rules obtained by a circular permutation of the indices.

We will also make use of the spin lowering and raising operators;

σ+ = |+〉〈−| = σx + iσy2

=

(0 10 0

); σ− = |−〉〈+| = σ†+ =

σx − iσy2

=

(0 01 0

), (2.48)

whose square is zero. They do not commute with σz and we have:

[σz, σ±] = ±2σ± . (2.49)

The most general observable (within a dimensional factor), σu, corresponds to the measurementof the spin component along an arbitrary direction defined by the unit vector u and the associatespherical coordinates angles θ and φ: u = (sin θ cosφ, sin θ sinφ, cos θ). It reads in matrix form:

σu =

(cos θ sin θe−iφ

sin θeiφ − cos θ

), (2.50)

and obviously coincides with the three first Pauli matrices when u is along one of the axes. Italso squares to unity and has thus the eigenvalues ±1 like all Pauli matrices. The correspondingeigenvectors are, within a global phase choice:

|+u〉 = |0u〉 = cosθ

2|+〉+ sin

θ

2eiφ |−〉 (2.51)

|−u〉 = |1u〉 = − sinθ

2e−iφ |+〉+ cos

θ

2|−〉 , (2.52)

where we have recalled the quantum logic notations.

When u spans all the directions, |+u〉 spans all accessible normalized states in the Hilbert space,within a global irrelevant phase. In fact, the most general state of a spin-1/2 is defined by two complexnumber, i.e. 4 real parameters. Normalization and the possibility to change at will the global phasemake two of these parameters irrelevant. Only two are free, and we can use θ and φ. There is thus abijection between the points on a unit sphere, the Bloch sphere, and the state of a spin-1/2 (see Fig.2.1).

A rotation on the Bloch sphere by an angle θ around the axis defined by v maps a state ontoanother one. The corresponding unitary operator can be expressed as:

Rv(θ) = e−i(θ/2)σv = cosθ

211− i sin

θ

2σv , (2.53)

where we made use for the rightmost part that the Pauli matrices square to unity. For instance, arotation by an angle θ around the uz axis is represented by the matrix:

Rz(θ) =

(e−iθ/2 0

0 eiθ/2

), (2.54)

with the immediate and intuitive consequence that Rz(π/2) |+x〉 = |+y〉. Note that Rv(2π) = −11, aremarkable feature of spin-1/2 systems.


|0ñ

X

Y

Z

f

|1ñ

|0Xñ

q

|0uñ

u

|0Yñ

Figure 2.1: Representation of the Hilbert space of a two-level system on the Bloch sphere. Using the quantum logics

notations, the |0〉 and |1〉 states correspond to the ‘north’ and ‘south’ poles respectively. The |0x〉 and |0y〉 states are

mirrored on the ‘equator’ of the sphere, at its intersection with the x and y axes. The most general spin state corresponds

to a point on the sphere of angular coordinates θ and φ.


Atomic Hamiltonian and observables

With these notations, and choosing the energy origin midway between levels |e〉 and |g〉, the atomicHamiltonian is:

H0 =hωeg

2σz . (2.55)

The dipole operator D = qR has no average value in the |e〉 or |g〉 state (R has odd parity and theatomic levels have a well-defined parity). The matrix representing D in the atomic Hilbert space thusreduces to non-diagonal elements (which are vectors in the real space):

D =

(0 dd∗ 0

), (2.56)

where d is a vector with a priori complex components describing the polarization coupled to thetransition (linear, circular or elliptical in the general case). In order to keep things simple, we assumethat d is real (corresponding, to the coupling of the transition to linearly polarized light in an arbitrarydirection). The dipole thus writes

D = dσx = d(σ+ + σ−) , (2.57)

a superpositions of the atomic raising and lowering operators as we could have expected.

The incoming wave has a field E(0, t) = E0 cos(ωt+ ϕ) at the position of the atom, where E0 is areal vector amplitude. We note the complex amplitude of this field as

E1 = E0e−iϕ . (2.58)

The atom-field coupling Hamiltonian is thus, in the D ·E representation:

H1 = −d ·E0 cos(ωt+ ϕ)σx . (2.59)

Defining the “Rabi frequency” Ω by:

Ω =d ·E0

h, (2.60)

we get:

H1 = −hΩ cos(ωt+ ϕ)σx . (2.61)

2.3.2 Rabi precession

Let us now study the evolution of the atomic system under the complete H0 +H1 Hamiltonian. Thefirst step is to get rid of the explicit time dependence of the electric field at frequency ω. For thispurpose, we first switch to an interaction representation with respect to the Hamiltonian H ′0 = hωσz/2.Note that this Hamiltonian differs from the atomic one by h∆σz/2, with

∆ = ωeg − ω , (2.62)

being the angular frequency difference between the atomic transition and the field. With these nota-tions, the full Hamiltonian is:

H = H ′0 +h∆

2σz +H1 . (2.63)

The interaction representation is defined by the evolution operator U ′0 = exp(−iH ′0t/h). The new

Hamiltonian is H = U ′†0H1U′0. The σz part of H1 is unchanged, since it commutes with the evolution

operator. We only have thus to compute the new spin raising and lowering operators σ± = U ′†0σ±U′0.


Using the Baker-Hausdorff lemma:

eBAe−B = A+ [B,A] +1

2![B, [B,A]] + . . . (2.64)

and expanding the commutators of σz (proportional to B) with those of σ+ = A, we get

σ+ = σ+ + iωtσ+ + (iωt)2σ+ + . . . = eiωtσ+ , (2.65)

a remarkably simple result. By a mere hermitian conjugation, we get:

σ− = e−iωtσ− , (2.66)

The interaction representation Hamiltonian is thus

H =h∆

2σz −

hΩ

2

(ei(ωt+ϕ) + e−i(ωt+ϕ)

) (eiωtσ+ + e−iωtσ−

). (2.67)

Embarrassingly, it still contains a time dependence. However, when expanding the product of expo-nentials, we see that we get two constant terms and two terms evolving at the frequencies ±2ω. Wecan thus perform a secular approximation, that will be valid as long as the Rabi frequency Ω, describ-ing the atom-field coupling, is much smaller than the field frequency, close to the atomic transitionone. We merely neglect the fast oscillating terms that, intuitively, average to zero in the Schrodingerequation. This approximation, that we will meet in a variety of contexts, is also called the “rotatingwave approximation” (RWA). The name derives from the Nuclear Magnetic Resonance context. Aspin rotates in a constant magnetic field at the Larmor frequency. A linearly polarized transversemagnetic field at the same frequency can induce a flip of the spin, but only the circularly polarizedcomponent rotating together with the spin is efficient in producing the effect. The other can be merelyneglected4.

With the RWA, we finally arrive at a time-independent Hamiltonian:

H =h∆

2σz −

hΩ

2

(σ+e

−iϕ + σ−eiϕ)

=h∆

2σz −

hΩ

2(σx cosϕ+ σy sinϕ) . (2.68)

From the second form of the total Hamiltonian, we see clearly that it is proportional to the spinoperator σn is a well defined direction:

H =hΩ′

2σn , (2.69)

with

n =∆uz − Ω cosϕux − Ω sinϕuy

Ω′, (2.70)

and

Ω′ =√

Ω2 + ∆2 . (2.71)

The evolution operator U(t) for a time interval t is thus

U(t) = e−i(Ω′t/2)σn = Rn(θ) , (2.72)

with

θ = Ω′t . (2.73)

On the Bloch sphere, the evolution is thus simply a rotation around n at the angular frequency Ω′, aquite simple result.

4In a more refined approximation, it induces a slight shift of the spin’s Larmor frequency.


When the detuning ∆ is much larger than the Rabi frequency Ω, the rotation axis coincides nearlywith the z axis, and the rotation frequency Ω′ with the detuning. If we start in one of the energyeigenstates |e〉 or |g〉, we remain there. A widely non resonant field cannot induce an atomic transfer,a quite intuitive result.

When ∆ = 0, the rotation frequency coincides with the Rabi frequency and the rotation axis liesin the equatorial plane of the Bloch sphere. The Bloch vector rotates on a great circle going throughthe two poles. When starting in |g〉, for instance, the probability for finding the atom in |e〉 at timet, pe(t), oscillates between zero and one according to:

pe(t) =1− cos(Ωt)

2. (2.74)

This is the Rabi oscillation phenomenon, discovered by I.I. Rabi, immediately after WWII, in one ofthe first atomic radio-frequency spectroscopy experiments.

The state transformations have simple expressions for well-chosen interaction times. Let us considerfor instance a so-called ‘π/2 pulse’, i.e. t = π/2Ω. The rotation of the Bloch vector is by π/2 aroundthe vector:

n = − cosϕux − sinϕuy , (2.75)

at an angle π+ϕ with the Ox axis in the equatorial plane of the Bloch sphere. The evolution operatoris thus the rotation:

Rn(π/2) =1√2

(11− iσn) =1√2

(1 ie−iϕ

ieiϕ 1

). (2.76)

It performs the state transformations:

|g〉 −→ 1√2

(|g〉+ ie−iϕ |e〉

)|e〉 −→ 1√

2

(|e〉+ ieiϕ |g〉

). (2.77)

This transformation prepares a superposition of |g〉 and |e〉 with equal weights and a phase defined bythat of the incoming field.

For Ωt = π (π-pulse), we perform, within phases, an exchange of the |e〉 and |g〉 level. Finally, forΩt = 2π (2π pulse), we leave the atomic state unchanged, besides the global − sign associated to a2π rotation of a spin-1/2.

When ∆ and Ω are comparable, the rotation is around an axis making a non-trivial angle α (givenby tanα = Ω′/∆) with the downwards z axis and the rotation around this axis is faster than inthe resonant case. It does not mean that the incident wave is more efficient in inducing transitions.Starting from |g〉, we cannot reach exactly state |e〉 whatever the interaction time.

The Bloch vector in its rotation generates a cone with an opening angle α and its maximum anglewith respect to the downwards z axis is 2α. The maximum probability pe,m for finding the atom in|e〉 is thus sin2 α and we get:

pe,m =Ω2

Ω2 + ∆2. (2.78)

The maximum transfer probability has a Lorentzian dependence versus the detuning ∆. This stressesthe resonant nature of the interaction. Most spectroscopic methods are based on it, either opticalspectroscopy or radiofrequency extensions, such as EPR or RMN. The width of the resonance isentirely determined by the Rabi frequency Ω and can, in principle, be made arbitrarily small.

To get an appreciable transfer from |g〉 to |e〉 at resonance, we need to perform a π pulse (Ωt = π).For a given interrogation time τ , this leads to Ω = π/τ and thus to a resonance width of the order ofπ/τ . We recover here a very simple Fourier analysis result: it is impossible to determine a frequencywith a precision larger than π/τ for a given interrogation time τ . Translated in terms of the energy of


the transition, hωeg, this remark boils down to the time-energy Heisenberg uncertainty relations. Ofcourse, in real life, relaxation process contribute to the linewidth and this optimal cannot be reachedfor long interaction times.

2.3.3 Ramsey separated oscillatory fields method

In order to perform a high-resolution spectroscopy experiment, the previous results suggest to apply along interrogation pulse to the system. This method is nevertheless often quite impractical, particularlywhen the atoms are moving at the typical velocities of room-temperature, as in most atomic clocks.Itrequires a very large atom-field interaction zone and puts severe constraints on the homogeneity ofthe probe field.

In 1949, N. Ramsey proposed a method that circumvents most of these difficulties. It has beeninstrumental in the development of high-resolution atomic spectroscopy and of atomic clocks. Theprinciple of the method is to divide a π Rabi pulse into two identical π/2 pulses, with a very shortduration τ , separated by a long time interval T of free atomic evolution. Each pulse alone would resultin a low resolution but their combination results in a spectroscopic resolution only limited by the totalinterrogation time T .

Principle of the method

We will assume that the π/2 pulses are identical and are obtained with a driving field phase ϕ = −π/2,Since their duration is very short, they are nearly resonant over a wide frequency range and we canthus assume ∆ = 0 to treat them. These “Ramsey pulses” induce thus, using Eq.(2.77), the statetransformations:

|e〉 −→ 1√2

(|e〉+ |g〉) (2.79)

|g〉 −→ 1√2

(− |e〉+ |g〉) (2.80)

Let us now follow the atomic state through time, assuming that we initially start in the lower state|g〉. After the first pulse (duration τ), the state is |Ψ(τ)〉 = (1/

√2)(− |e〉 + |g〉). During the time T ,

the atomic state evolves freely. For this long duration, we cannot neglect the detuning ∆ as is the caseduring the short Ramsey pulses. The atom evolves during this time interval under the Hamiltonian(h∆/2)σz, producing the state transformations |e〉 → exp(−iΦ/2) |e〉 and |g〉 → exp(iΦ/2) |g〉, withΦ = ∆t. Within an irrelevant global phase, the state at time T + τ ≈ T is thus

|Ψ(T )〉 =1√2

(− |e〉+ eiΦ |g〉

)(2.81)

The final π/2 pulse performs the same state transformations as the first. We are thus left with thefinal state:

|Ψf 〉 = −1

2

[(1 + eiΦ

)|e〉+

(1− eiΦ

)|g〉)

(2.82)

The final probability for finding the atom in the upper state |e〉 is thus:

pe =1

4

(1 + eiΦ

)2=

1

2(1 + cos ∆T ) . (2.83)

It is oscillates between zero and one as a function of ∆, with a period 2π/T . The spectroscopicresolution on the measurement of the atomic frequency ωeg is thus of the order of 1/T , as if a singleπ pulse with duration T would have been applied on the atom. This clever trick makes it possi-ble to achieve interrogation-time limited resolution in all atomic clocks. It also allows for a precisemeasurement of perturbations acting on the atom during the free evolution time. The phase of the


pe oscillations is determined by the time integral of the atom-probe field detuning over time T andreflects any shift, even a transient one, acting on the atom in-between the two Ramsey pulses.

The Ramsey method can also be viewed as an atomic interference process. The first pulse createsan atomic state superposition, and the second mixes again the two atomic levels. This is reminiscentof a Mach-Zehnder light interferometer, in which a first beam-splitter coherently splits the incominglight into two beams, later recombined by a second beam-splitter. The final interference of the twobeams reflects any phase shift they undergo in their separated paths though the interferometer. Here,the paths are not separated in space, but in atomic states. There are two quantum paths leading theatom from |g〉 to |e〉, in which it makes the transition either during the first pulse or during the second.The Ramsey signal results from the quantum interference of these two paths. The names “Ramseyinterferometer” and “Ramsey fringes” are thus often used to describe the method.

Signal to noise ratio

We have qualitatively estimated the precision of the atomic frequency measurement to be of the orderof 1/T in the Ramsey interferometer. Let us make this argument more precise by considering in moredetails the detection statistical noise problem.

We will assume that we send N independent atoms through the interferometer. Let us call Ne thenumber of atom detected in |e〉, an obviously random quantity. The mean number of atoms detectedin |e〉 is

〈Ne〉 =N

2(1 + cos Φ) , (2.84)

with Φ = ∆T . The variance of Ne is given by:

∆2Ne = Npe(1− pe) =N

4sin2 Φ , (2.85)

and hence

∆Ne =

√N

2sin Φ . (2.86)

This variance, associated to the random nature of Ne is called “projection noise”.In order to assess the resolution of the method, we perform two measurements of N atoms, with

two different settings of the interrogation source frequency, corresponding to ∆ and ∆ + δ. Assumingthat the detuning increment δ is small, we get

〈Ne(∆ + δ)〉 = 〈Ne(∆)〉 − NT

2δ sin ∆T . (2.87)

We will be able to distinguish the two values of the detunings provided the variation of the mean isgreater than the square root of the variance multipled by

√2 (the statistical errors on two independent

measurements add quadratically), i.e. provided

NT

2δ sin ∆T >

√2

sin ∆T

2

√N , (2.88)

or

δ >

√2

T√N

. (2.89)

We thus get a more precise expression of the spectroscopic sensitivity of the method. The firstobservation is that it is independent upon the Ramsey interferometer phase Φ. When Ne is maximumor minimum (top or bottom of the Ramsey fringes signal), the noise on it is minimal but the derivativeof the signal versus the detuning is also minimal. At a mid-fringe setting, where pe = 1/2, thesensitivity to the detuning variation is optimal, but the statistical noise is also high. We also observethat the sensitivity ranges like

√N , a typical feature for N independent measurements of the same

quantity. Using more sophisticated entangled atomic states makes it possible in principle to reach amuch higher sensitivity, ranging like N (Heisenberg limit).


2.4 Relaxing atom and resonant field

The model developed in the previous section is quite unrealistic. We have assumed that the atomicstates are stationary, including the upper state |e〉 of the transition. We should at least take intoaccount the spontaneous emission from this level. In realistic situations, all kinds of random fieldsmay also act on the atoms (fluctuations of the local electric or magnetic fields, inducing random Starkor Zeemann effects). These fields may spoil the atomic levels coherence and will certainly contributeto a limitation of the spectroscopic resolution.

Introducing these effects will lead us to the famous Optical Bloch Equations describing the atom-field interaction is a realistic way in a wide variety of situations. We will extensively describe someof their main applications. This Section will also be the opportunity to review first the description of‘open’ quantum systems coupled to a complex environment.

2.4.1 Description of the atomic relaxation

We consider the rather general problem of a quantum system S (the atom here) coupled to an en-vironment E . By ‘environment’ we mean a large collection of systems or degrees of freedom, with acontinuous and wide spectrum of characteristic frequencies, in thermal equilibrium at some tempera-ture T (possibly zero).

The environment and the system are together a closed quantum system, described by a pure state|ΨSE〉. The environment being fantastically complex, we are not interested in this state but onlyin the reduced system state described by the density operator ρS , obtained by tracing the projector|ΨSE〉〈ΨSE | over the environment.

We are thus seeking an equation of evolution for ρs (that we will now note ρ since there is noambiguity). The standard approach5 starts from a model of the environment. The equation of motionof ρ is obtained by a complex procedure (some 20 pages), involving two stages of approximation. TheBorn approximation amounts to treating the system-environment interaction to the lowest significantorder. Physically it means that any excitation which the system looses in the environment will be lostforever, and never comes back in the system. This is clearly highly valid for thermodynamics reservoirs,made of infinitely many degrees of freedom. The Markov approximation neglects the memory of thereservoir: it assumes that all its observables have, in the Heisenberg picture, an extremely smallcorrelation time τc. This is also very well verified for all reasonable reservoir models. It is particularlythe case for spontaneous emission. The reservoir is then the continuum of all optical modes of thefield, with a width of the order of the optical frequencies themselves, and thus a correlation time in the10−15 s range. The equation of motion of ρ is then obtained. Its form finally does not depend aboutthe detailed structure of the reservoir. Only the numerical values of the relaxation times involved inthe dynamics do depend a priori on the details of the model. This is not a serious concern, though,since these times can also be measured independently in dedicated experiments.

We will follow here a much more intuitive and direct path. It will provide us easily with theequation of motion of ρ by considering the “quantum jumps” that the system may undergo under theinfluence of its coupling with the environment6

Kraus operators

We thus consider a short time interval τ and consider the transformation of the system’s densitymatrix from time t to time t+ τ :

ρ(t) −→ ρ(t+ τ) . (2.90)

5See for instance S. Barnett’s Theoretical methods in Quantum Physics, Oxford University Press.6More details on this approach are in Haroche and Raimond Exploring the quantum and in the course by Preskill on

quantum information, freely available on the Web at http://www.theory.caltech.edu/people/preskill/ph229/#lecture.

2.4. RELAXING ATOM AND RESONANT FIELD 37

The time interval τ is chosen to be small at the scale of the system’s dynamics (i.e. small compared toall characteristic frequencies and relaxation times), so that the modification of ρ is only incremental.On the other hand, we chose it much longer than the correlation time of the environment, τ τc, sothat there are no coherent effects in the system-reservoir interaction. For spontaneous emission, forinstance, the dynamics occurs at the time scale of the spontaneous emission lifetime 1/Γ ' 10−9 s andτc ' 10−15. There is thus a wide possible range for the time increment τ . Since the environment is avery large system, with a very short memory time, we can assume that it is always disentangled withthe system and in some reference state, for instance reflecting its thermal equilibrium at temperatureT .

The transformation (2.90) is an example of a “quantum map” operation: L(ρ(t)) = ρ(T + τ). Thisquantum map operation should obey some important mathematical rules:

• It should be a linear operation, i.e. a super-operator acting in the Hilbert space of operators.This operator space is of dimension N2

S if the system’s Hilbert space of S is of dimension NS .

• It should preserve the unit trace of the density operator, as well as its positivity (a densityoperator does not have any negative eigenvalue).

• It should be “completely positive”. If, at a time t, S is entangled with another system S ′, theapplication of the quantum map L to S should lead to a completely positive density operatorfor the entangled state of S and S ′ at time t + τ . Let us only mention at this stage that thereare quantum maps, which obey the first two properties but which are not completely positive(the partial transpose is an example).

When L checks these three properties, it is possible to show (the mathematical derivation is abit longish and will not be reproduced here) that there are at most N2

S operators Mµ acting on thesystem’s Hilbert space (not necessarily Hermitian or unitary) such that the map can be written in theKraus form:

L(ρ) =∑µ

MµρM†µ , (2.91)

with the normalization condition ∑µ

M †µMµ = 11 . (2.92)

The Kraus operators Mµs are not unique. Any linear unitary transformation mixing them leavesthe quantum map unchanged. There are thus many ways in which the same quantum map can bedescribed. This is a very simple and powerful mathematical result. It also brings a considerablesimplification, since all the action of the formidably complex environment E can be “wrapped” in theMµ operators.

Note that the evolution of S under the Hamiltonian H can be cast in this form also, since ρ(t+τ) =U(τ)ρU †(τ), where U is the unitary evolution operator. A generalized measurement of S cannot becast in this form. It is described by at most N2

S arbitrary operators Oµ but the map is non linear:

ρ −→OµρO

†µ

TrρO†µOµ, (2.93)

where the denominator is simply the probability pµ for getting µ as the result of the measurement. Ifwe do not take notice of the measurement result, we should average the final density matrix over allpossible results weighed by their occurrence probabilities. The mapping due to an unread measurementthus writes:

ρ −→∑µ

OµρO†µ , (2.94)

and appears in the Kraus form. The Kraus process encapsulates thus most quantum processes.


Lindblad equation

We can write the quantum map L leading from the density operator at time t to that at time t + τ ,but we are obviously more interested in a differential “master” equation for the evolution of ρ. Wewill now show that it can be straightforwardly derived from the Kraus operators.

Since the environment is at all times close to its thermal equilibrium and does not take part inthe dynamics, it is natural to assume that the Kraus operators Mµ do not depend upon time. They,however, depend clearly upon the tiny time interval τ .

If the density operator evolves smoothly, we have:

ρ(t+ τ) =∑µ

MµρM†µ ≈ ρ(t) +

dρ

dtτ . (2.95)

One and only one of the Mµs is thus of the order of unity and all others must then be of order√dτ .

We will write, without loss of generality:

M0 = 11− iKτ (2.96)

Mµ =√τLµ forµ 6= 0 , (2.97)

Note that M0 might a priori include corrections at order√τ . These corrections can nevertheless be

removed from M0 and incorporated in one of the Mµs.The operator K, which has no particular a priori mathematical properties, can be split in its

hermitian and anti-hermitian parts:

K =H

h− iJ , (2.98)

where

H =h

2

(K +K†

)(2.99)

J =i

2

(K −K†

)(2.100)

are both hermitian. With these notations:

M0 = 11− iτ

hH − Jτ . (2.101)

To first order in τ , we now have:

M0ρM†0 = ρ− iτ

h[H, ρ]− τ [J, ρ]+ , (2.102)

where [J, ρ]+ = Jρ+ ρJ is an anti-commutator. The commutator in the r.h.s. of the above equationclearly evokes the Hamiltonian evolution of the density operator. We also have to the same order:

M †0M0 = 11− 2Jτ . (2.103)

The normalization condition∑µ M

†µMµ = 11 thus writes simply:

J =1

2

∑µ 6=0

L†µLµ . (2.104)

Injecting these expressions in the Kraus map, we finally get the density matrix master equation inthe so-called “Lindblad form”:

dρ

dt= − i

h[H, ρ] +

∑µ6=0

(LµρL

†µ −

1

2L†µLµρ−

1

2ρL†µLµ

). (2.105)

Note that the last two terms in the r.h.s. sum are included to preserve the trace of the density matrix.The dynamics is in fact mostly governed by the first term in the sum, Since the Mµs are not uniquelydefined, there is also a considerable freedom in the way we can write the same master equation.


2.4.2 Quantum jumps

Let us now get an insight into the physical interpretation of the Lµ operators appearing in the Lindbladmaster equation. For this purpose, let us consider a simple situation, in which the initial density matrixis a pure state ρ(0) = |Ψ〉〈Ψ| and let us assume that there is no Hamiltonian evolution (or that we usea proper interaction representation cancelling H). After a small time interval τ , the un-normalizeddensity operator7 is:

ρ(τ) = |Ψ〉〈Ψ|+ τ∑µ

(Lµ |Ψ〉)(〈Ψ|L†µ

). (2.106)

Either, with a probability of order one, the quantum state of the system is unchanged, or, withprobabilities of order one in τ , the system undergoes a large evolution described by one of the Lµoperators.

We are thus lead to interpret the Lµs as ‘jump operators’ which describe a random (perhaps large)evolution of the system which suddenly (at the time scale of the evolution) changes under the influenceof the environment. Of course the density operator evolves slowly, since these quantum jumps occurwith a small probability, at the first order in τ .

The interest of the method is that, in many cases, the nature of the quantum jumps can be guessedfrom the mere nature of the system. Anticipating a lot, we for instance guess that a field cavity lossesare represented by a jump operator proportional to the annihilation operator a, describing the lossof a single photon. We also guess that, for an atom, the jump operator should be proportional tothe lowering spin operator σ−. The proportionality factors (the actual relaxation rates) cannot beobtained in this way, but they can at least be measured. We have thus a very simple path to guide usin writing the proper Lindblad master equation for any system.

Once again, the jump operators are not uniquely defined. The same relaxation process can bemodeled in different ways, different ‘unravelings’ of the master equation. In some situations, the natureof the system privileges one of these unravelings. For instance, with an emitting atom completelysurrounded by a photo-detector array. The quantum jump then corresponds to a click of one detector.Different unravelings may then correspond to different ways of monitoring the environment, in thiscase to different detectors (photon counters, homodyne recievers...). In other situations, the quantumjumps are an abstract representation of the system+environment evolution.

Even when the environment is not explicitly monitored, one may imagine that it is done. Wethen imagine we have full information about which quantum jump occurs when. The system is thus,at any time, in a pure state, which undergoes a stochastic trajectory in the Hilbert space, made upof continuous Hamiltonian evolutions interleaved with sudden quantum jumps. However, since weonly imagine the information is available, we should describe the evolution of the density operator byaveraging the system evolution over all possible trajectories.

The ‘environment simulator’ concept provides a simple recipe to perform this averaging. Weintroduce a system B, much simpler than the formidably complex environment E and couple it to Sin such a way that the reduced dynamics of S is exactly ruled by the Lindblad equation (2.105). Forthis, it is enough to check that S undergoes the same quantum map during the time interval τ as withits interaction with E . We also assume that B is prepared at the beginning of all time intervals τ inthe same reference state |0〉, reflecting the fact that in the real situation E never gets modified by S.

We assume that S and B undergo an Hamiltonian evolution during τ , resulting in the globalunitary evolution operator USB such that:

USB |Ψ〉 ⊗ |0〉 = M0 |Ψ〉 ⊗ |0〉+∑µ

(Mµ |Ψ〉)⊗ |µ〉 , (2.107)

where the first ket refers to S and the second to B, and Mµ =√τLµ are the Kraus operators. The

|µ〉 states of B are assumed to be orthogonal. Defining an Hamiltonian leading to this action of the

7We do not take into account the last two terms in the r.h.s. sum of Eq.(2.105).


evolution operator is always possible, provided the dimension of the Hilbert space of B is at leastas large as the number of Kraus operators, at most N2

S . The ‘environment simulator’ is thus muchsimpler than the actual environment.

At the end of the time interval τ an unspecified observer performs a measurement of an observableOB having the |µ〉s as non degenerate eigenstates, with µ as the eigenvalue. Applying the measurementpostulates, we immediately find that

• With a probability p0 = 〈Ψ|M †0M0 |Ψ〉 = Tr(ρM †0M0) = 1− τ∑µ6=0 Tr(ρL†µLµ) = 1−

∑µ 6=0 pµ,

the result is 0, no jump occurs and the system’s state is finally left in the normalized state

M0 |Ψ〉√p

0

=1− iHτ/h− Jτ

√p

0

|Ψ〉 . (2.108)

Note that the state vector evolution involves an Hamiltonian part, iHτ/h, (resulting alonein a unitary evolution) and a part reflecting normalization whose evolution operator is non-unitary. This no-jump evolution can be interpreted as resulting from the non-hermitian effectiveHamiltonian:

Heff = H − ihJ . (2.109)

• With a probability pµ = τTr(ρL†µLµ), the result is µ and the system’s state is accordingly

projected ontoMµ|Ψ〉√

pµ=

Lµ|Ψ〉√pµ/τ

. Note that the mere definition of the probabilities ensures that∑µ=0 pµ = 1.

Of course, the information provided by the unspecified observer of the environment simulator isnot available (in most cases, we do not have any information about what happens in the environment).We have thus to assume that the measurement at the end of the time interval τ is unread. This resultsin a quantum map acting on the system state that is precisely expressed by

∑µ MµρM

†µ, identical to

the environment-induced quantum map. The evolution of the system coupled to B is thus the sameas when it is coupled to E .

The different unravelings of the master equation obviously correspond to the measurement ofanother observable in the environment simulator B, whose eigenvectors are linear combinations of the|µ〉s, defined by a unitary basis change matrix.

2.4.3 Quantum Monte Carlo

The quantum jump approach also leads to a very efficient method for computing numerically theevolution of the system: the ‘quantum Monte Carlo’ method.

Computing a solution to the Lindblad master equation is complex, because the density operatorinvolves a number of parameters of order N2

S . Direct integration, straightforwardly feasible for anHilbert space dimension of a few hundreds, becomes nearly intractable for much larger Hilbert spaces.This is a very strong limitation for many practical problems, such as those involved in cold atomsphysics.

The above discussion nevertheless provides us with a useful approach. Instead of assuming thatthe measurement performed by the unspecified observer of B is unread, we will take it as a projectivemeasurement whose result is known to us. In this case, the system initially in a pure state is still in apure state at the end of the time interval τ and thus remains forever in a pure state. We can compute,by throwing the dice to determine the result of each measurement, the time evolution of this purestate over a particular ‘trajectory’. Of course, a single trajectory does not tell the whole story. Itcan be shown easily that the density operator obtained by averaging the projector on this pure stateover many realisations of individual trajectories, determined by different dice throwing for each timeinterval, is the solution of the Lindblad equation with the initial pure state as initial condition. When


the initial state is not a pure state, we can obviously diagonalize the density operator, evolve each ofits pure state components and finally average the result.

The quantum Monte Carlo algorithm is thus quite simple

• Choose a time interval τ short on the time scale of the relaxation, but long compared to thereservoir correlation time.

• Initialize the state to its initial value, or to a randomly chosen eigenstate of the density operator(if the initial state is not pure).

• For each time interval τ over the total evolution time evolve the quantum state |Ψ〉 accordingto the rules:

– Compute the probabilities pµ = τ 〈Ψ|L†µLµ |Ψ〉 and p0 = 1−∑µ 6=0 pµ.

– Use a (good) random number generator to decide upon the result of the measurement ofthe environment simulator B.

– If the result of the measurement is zero, evolve |Ψ〉 with

|Ψ〉 −→ 1− iHτ/h− Jτ√p0

|Ψ〉 . (2.110)

– If the result of the measurement is µ 6= 0, evolve |Ψ〉 by the jump operator Lµ:

|Ψ〉 −→ Lµ√〈Ψ|L†µlµ |Ψ〉

|Ψ〉 =Lµ√pµ/τ

|Ψ〉 . (2.111)

• Repeat the procedure for a large number (a few hundreds is most often enough) of trajectories

• Average the projectors on the states and obtain an estimate of the time-dependent densityoperator solution of the Lindblad equation

ρ(t) = |Ψ(t)〉〈Ψ(t)| . (2.112)

This procedure allows to compute only state vectors with NS parameters. It is thus much fasterthan the direct integration of the Lindblad equation, as soon as the Hilbert space dimension is greaterthan a few hundreds. It is now widely used in a variety of contexts.

Spontaneous emission

Let us now turn to the focus of this Chapter and examine the master equation describing the spon-taneous emission by a single two-level atom in which |g〉 is the ground state. We assume here azero-temperature environment. This is an excellent approximation for optical transitions at roomtemperature, since the average thermal energy, kbT (about 1/40 eV), is much smaller than the tran-sition energy (at least 1 eV).

It is easy to identify the jump operators. Let us for that consider a simple environment simulator,that corresponds to an experiment in which the atom would be totally enclosed in unit detectionefficiency photo-counters. Then, either nothing happens in a time interval τ (the counters remaininactive), or one of the counters clicks. This reveals the emission of a photon while the atom makesthe transition from |e〉 to |g〉, described by the atomic lowering operator σ−. We can thus guess thatthe emission can be modelled by a single jump operator L:

L =√

Γσ− , (2.113)


where Γ is the spontaneous emission rate. The Lindblad master equation thus writes (we assume thatthe Hamiltonian term cancels or that we use a proper interaction representation):

dρ

dt= Γ

(σ−ρσ+ −

1

2σ+σ−ρ−

1

2ρσ+σ−

), (2.114)

where we have used σ†− = σ+ (note also that σ+σ− = |e〉〈e|).Writing ρ in matrix form as:

ρ =

(ρee ρegρge ρgg

), (2.115)

(the first index in the subscripts being the line index as usual), where ρee + ρgg = 1 and ρeg = ρ∗ge, weget the only two relevant equations:

dρeedt

= −Γρee , (2.116)

dρegdt

= −Γ

2ρeg . (2.117)

We thus obtain the simple result that the population of level |e〉, ρee, is exponentially damped witha time constant T1 = 1/Γ, whereas the coherence between the levels is damped with a time constant2T1. Note that at a finite environment temperature, we should add a σ+ jump operator, describingthe absorption of a thermal photon by the atom, making in the process a transition from |g〉 to |e〉.

Let us analyse the spontaneous emission process in terms of Monte Carlo trajectories. The atomis assumed to be initially in |e〉, At each time step, either there is no jump or there is one. In the firstcase, only the normalization operator J ∝ σ+σ− acts on the wavefunction. Since |e〉 is an eigenvectorof this operator, the atomic state does not change when the no-jump option is selected, a ratherintuitive result. When a jump occurs, it is described by σ− which turns the state into |g〉. From thistime on, no jump can act. It is easy to show that the jump probability cancels when the atom is in|g〉, a fortunate fact to keep normalization since σ− |g〉 = 0.

A quantum Monte Carlo trajectory is thus a ‘telegraphic’ signal, where the atom remains in |e〉for some time and suddenly jumps into |g〉. The time of the jump is random (we leave to the readeras an exercise in probabilities to compute its statistical distribution). Of course, a single trajectorydoes not tell the whole story. We have to average many of these telegraphic trajectories. In this way,we recover the average exponential damping, solution of the Lindblad equation.

We can get a bit more insight by considering as initial state a superposition of |e〉 and |g〉: |Ψ0〉 =(1/√

2)(|e〉 + |g〉). Let us follow a trajectory. Before the jump (only one allowed of course), thewavefunction evolves due to the non-hermitian Hamiltonian Heff which writes here:

Heff = − ih2

Γσ+σ− = − ih2

Γ |e〉〈e| . (2.118)

Writing the state at time t as the un-normalized ket ce |e〉+ cg |g〉, a mere substitution leads to:

ihdcedt

= − ih2

Γce , (2.119)

whose solution isce(t) = e−Γt/2 . (2.120)

The normalized wavefunction is then

|Ψ(t)〉 =1

1 + e−ΓT

(e−Γt/2 |e〉+ |g〉

). (2.121)

When the quantum jump occurs, at a random time, the state is suddenly projected onto |g〉 andno further evolution takes part, since |g〉 is an eigenstate of Heff with eigenvalue 0.

2.5. OPTICAL BLOCH EQUATIONS 43

Now the state evolves even before the quantum jump, the probability for being in |e〉 being ex-ponentially damped with the rate Γ. We can understand that in terms of the measurements in theenvironment simulator. When no photon is detected for a while, it provides information on the systemstate. It tells that it is less likely that the atom is excited, since it does not emit. It is importantto recognize here that an experiment with a negative result (no photon counted) does indeed changethe state of the system, not as brutally though as a quantum jump. In the limit where no photonhas been observed for a very long time, it is clear that the system must be in state |g〉. Observing nophoton at any time is performing a projective measurement of the atomic state.

Phase damping

Spontaneous emission is not the only relaxation process than can affect an atom. It can also besubmitted to random fields that modify in an uncontrolled way its resonant frequency. In an intuitivepicture, these fields will not affect the states population, but they will provoke random dephasings ofthe coherence between |e〉 and |g〉 and finally damp these coherences. Let us add the correspondingterms in the Lindblad equation.

The extreme event on the phase of the atomic coherence is a π phase shift, represented by the σzoperator. We will thus use

√γ/2σz as the jump operator, where γ = 1/T2 is a ‘transverse’ relaxation

rate and T2 the associated transverse relaxation time constant. We call them transverse becausethey describe a process affecting only the transverse components of the Bloch vector representing thedensity operator. By opposition we will call T1 a ‘longitudinal’ relaxation time.

After a little bit of algebra, that we leave to the reader as an exercise on Pauli matrices, we obtainthe master equations for the populations, which remain constant as expected, and for the coherenceρeg, which is exponentially damped with a rate γ.

Adding the spontaneous emission and the phase damping in the same Lindblad equation, we getthe complete master equation for the density matrix elements:

dρeedt

= −Γρee (2.122)

dρegdt

= −Γ

2ρeg − γρeg = −γ′ρeg , (2.123)

where we define the total relaxation rate of the coherence by:

γ′ = γ +Γ

2. (2.124)

At finite temperature, the Lindblad master equation would include all the operators σz and σ±.They are, with the trivial unity, a basis of all possible spin operators. There is thus no new relaxationprocess that we could add by making the Lindblad equation more complex.

2.5 Optical Bloch equations

We now turn to the core of this Chapter on the semi-classical approach, by writing the Optical Blochequations that describe the evolution of the atomic field operator under the joint influence of anexternal classical driving field and of a general (zero temperature) relaxation process, described bythe transverse and longitudinal relaxation times introduced in the previous Section. We first writethe equations, revisit in this frame the Rabi oscillations phenomenon. We examine it in two limitingcases (strong transverse relaxation or stochastic driving field). In both cases we will recover, mutatismutandis, the phenomenological Einstein’s coefficients model and get an insight into their validityrange.


2.5.1 The equations

OBE

Let us recall here that the Hamiltonian (2.68) describing the action of the classical field is, in theinteraction representation with respect to the field frequency:

H =h∆

2σz −

hΩ

2

(σ+e

−iϕ + σ−eiϕ), (2.125)

with Ω = dE0/h and ∆ = ωeg − ω being the angular frequency difference between the atom and thefield (we assume throughout this Section that d is real). We omit here the tildes for the interactionrepresentation that we will always use from now on. We will note E1 the complex amplitude of thedriving field:

E1 = E0e−iϕ . (2.126)

The Schrodinger equation describing the coherent evolution of ρ, ihdρ/dt = [H, ρ] leads, afterevaluation of the commutators, to the two coupled equations which together entirely determine theevolution of the density matrix:

dρeedt

= ΩIm(ρege

iϕ)

=d

hIm (ρegE

∗1) , (2.127)

and

dρegdt

= −i∆ρeg + iΩ

2e−iϕ(ρgg − ρee)

= −i∆ρeg − id

2hE1(ρee − ρgg) (2.128)

The electric field couples the populations (precisely the population inversion) and the coherences.We now add relaxation in the Lindblad form, taking into account both the longitudinal (sponta-

neous emission) and the transverse (dephasing) mechanisms. We must take care that we are now in aninteraction representation at the driving field frequency ω, which has changed σ± into σ± exp(±iωt)and leaved σz unchanged. However, all terms in the relaxation part of the Lindblad equation in-volve products of the form σ−σ+ which are left unchanged. The coupled equations of motion of thepopulation and coherences, the Optical Bloch Equations are thus:

dρeedt

=d

hIm (ρegE

∗1)− Γρee (2.129)

dρegdt

= −i∆ρeg − id

2hE1(ρee − ρgg)− γ′ρeg , (2.130)

where once again Γ = 1/T1 and γ′ = (1/2T1) + 1/T2.We now proceed to put these important equations under equivalent and equally useful forms.

Noting Ne = ρee and Ng = ρgg the populations of the states and introducing the complex dipoleamplitude D 8

D = 2dρeg , (2.131)

so that the average value of the hermitian dipole operator in state ρ is ReD, we put the OBE undera form frequently encountered in literature:

dNe

dt=

1

2hIm (DE∗1)− ΓNe (2.132)

dDdt

= −i∆D − γ′D − id2E1

h(Ne −Ng) (2.133)

8 D, should not be mistaken for the population inversion D = Ne −Ng introduced in the previous Chapter.


Let us introduce the Cartesian coordinates of the Bloch vector r = (x, y, z) (often noted r =(u, v, w) in the NMR context) defined by:

ρ =1 + r · σ

2(2.134)

or equivalently by:

ρ =1

2

(1 + z x− iyx+ iy 1− z

), (2.135)

i.e. byx = 2Re ρeg y = −2Im ρeg z = 2ρee − 1 . (2.136)

with the notation E1 = Ex + iEy (where both Ex and Ey are real) we finally arrive at the three realcoupled equations:

dz

dt= −d

h(xEy + yEx)− Γ(1 + z) (2.137)

dx

dt= −∆y +

d

hzEy − γ′x (2.138)

dy

dt= +∆x+

d

hzEx − γ′y . (2.139)

The interpretation of these equations is straightforward. The detuning ∆ results in a rotation of theBloch vector around the z axis. The Ex component of the electric field induces a rotation of the Blochvector around the x axis, coupling the y and z components, and the Ey component a simultaneousrotation around the y axis coupling x and z. Relaxation leads z to the −1 point corresponding to theatomic ground state, while the transverse relaxation damps the x and y coordinates.

Rabi oscillations Revisited

In this section, we revisit the Rabi oscillation phenomenon in the frame of the OBEs, includingrelaxation. In order to make the algebra a bit simpler without losing the main features, we will makethe following assumptions:

• The initial state is the ground state |g〉 corresponding to z = −1 and x = y = 0.

• The field is purely real: Ey = 0, Ex = +E0

• Atom and field are at resonance: ∆ = 0.

From these hypotheses, it is clear that x remains zero at any time and that the whole dynamicsis contained in the xOz plane. The corresponding equations of motion are, in terms of the Rabifrequency Ω = dE0/h:

dz

dt= −Ωy − Γ(1 + z) (2.140)

dy

dt= Ωz − γ′y . (2.141)

We can extract a single second-order differential equation for z from this system. We derive the firstwith respect to time, replace dy/dt using the second. We are left with an y term in the equation thatwe can eliminate by getting y as a function of z and dz/dt from the first equation. After a bit ofrearranging, we get finally:

d2z

dt2+ (Γ + γ′)

dz

dt+(Ω2 + γ′Γ

)z = −γ′Γ . (2.142)


This equation can be solved by standard methods and exhibits a transient behaviour [lasting atime of the order of 1/(Γ + γ′)] towards a steady state point, instead of the permanent Bloch vectorrotation obtained in the relaxation-free case. The final steady state point coordinates, zs and ys canbe obtained easily from the equations:

zs = − γ′Γ

Ω2 + γ′Γ, (2.143)

and

ys =Ω

γ′z = − ΩΓ

Ω2 + γ′Γ, (2.144)

Not surprisingly, we find ys = 0 and zs = −1 in the limit of a vanishing driving field Ω → 0. In theopposite limit, Ω → ∞, we get instead zs = ys = 0. The Bloch vector is in the centre of the sphere,corresponding to an equal weight mixture of |e〉 and |g〉. In an intuitive picture, a fast oscillating spinis in such a mixture of the two levels.

The transient regime is algebraically complicated. In order to get a simpler picture, let us assumethat we have no transverse relaxation (pure spontaneous emission) and, hence, that γ′ = Γ/2. Letus furthermore assume Ω Γ (strong driving of the transition). We can thus write the evolutionequation as:

d2z

dt2+

3Γ

2

dz

dt+ Ω2z = 0 , (2.145)

where we have neglected the constant term since, in this regime, the asymptotic value zs is nearlyzero. The solution, with the initial condition z(0) = −1 is:

z(t) = − cos(Ωt)e−3Γt/2 , (2.146)

an exponentially damped Rabi oscillation at the frequency Ω.We can get a qualitative insight into the nature of the Rabi oscillation damping by returning to the

quantum Monte Carlo trajectory picture. The non-hermitian part of the Hamiltonian can be neglectedas compared to the strong driving. The no-jump evolution is thus an unchanged Rabi oscillation atfrequency Ω starting from |g〉. Jumps occur in this case at an average rate Γ/2 since, on the average,the atom is in |e〉 half of the time. A strongly driven atom emits on the average one photon each 2T1,a simple result that will be useful later. After a jump, the atom is projected in the lower state |g〉 andthe Rabi oscillation restarts. However, its phase has changed, since the quantum jump never occurswhen the atom is in |g〉. The oscillation continues with this phase up to the next quantum jump, thatchanges again randomly the oscillation phase. An average over many trajectories thus results in aprogressive washing out of the oscillations.

Oscillator strength

Let us, mostly for the sake of completeness, briefly compare the results of this Section to those ofthe perturbative model introduced in Section 2.2, in the case where the atom is initially in state |g〉,driven by a widely detuned field ∆ Γ, γ′. Obviously, the atom remains at any time close to itsinitial state and we can assume that Ng = 1. The steady-state value of the complex dipole amplitudeD = 2dρge, where d = 〈e| z |g〉, is given by Eq. (2.133):

Ds =d2

h∆E1 =

q2| 〈e| z |g〉 |2

h(ωeg − ω)E1 . (2.147)

Since Ds is proportional to the complex electric field amplitude, this defines a quantum polarizabilityαQ by Ds = ε0αQE1. We get

αQ =q2

h(ωeg − ω)| 〈e| z |g〉 |2 . (2.148)


Comparing it to the classical polarizability in similar conditions:

αc =q2

2mε0ωeg

1

ωeg − ω, (2.149)

we get an expression of the ‘oscillator strength’ of the atomic transition:

f =2mωegh| 〈e| z |g〉 |2 , (2.150)

which coincides obviously with that obtained in Section 2.2.

2.5.2 Two limit cases: back to Einstein’s coefficients

We consider in this paragraph two limit cases in which the Einstein’s coefficients model can be directlyretrieved from the OBEs.

Strong transverse relaxation

The first case is that of a strong transverse relaxation. We thus assume that the damping rate γ′ ≈ γis very large compared to all other rates and frequencies appearing in the OBEs (2.132) and (2.133).In the evolution equation for D, we thus neglect dD/dt as compared to γ′D. This amounts to assumingthat D is at any time in its steady state value. The remaining equation is merely algebraic and wecan solve it after this ‘adiabatic elimination’ as:

D =i

γ′ + i∆

d2E1

h(Ng −Ne) , (2.151)

(note that this expression is valid in the steady state for all γ′ values – we will use it later in thiscontext). We then inject this expression in the equation of evolution of Ne (in which dNe/dt cannotbe neglected in front of ΓNe since Γ γ′):

dNe

dt= −ΓNe +

1

2hIm

[i

γ′ + i∆

d2E1

h(Ng −Ne)E

∗1

]

= −ΓNe +d2E2

0

2h2 (Ng −Ne)γ′

γ′2 + ∆2(2.152)

The squared field amplitude, E20 , is proportional to the energy density of the field, u. Since the field

is in principle strictly monochromatic, we cannot attribute to it a regular spectral energy density uν .We can nevertheless assume that the field has a finite, if small width and that we can make use of aspectral density of energy, uν , obviously proportional to E2

0 .We will assume furthermore that the field is resonant with the atomic transition (∆ = 0). We can

then rewrite the master equation for Ne as:

dNe

dt= −ΓNe +

d2E20

2h2γ′(Ng −Ne) = −ΓNe +

Ω2

2γ′(Ng −Ne) , (2.153)

and cast it straightforwardly in terms of the Einstein’s transition rates:

dNe

dt= AegNe + (BgeuνNg −BeguνNe) , (2.154)

with the evident correspondence Aeg = Γ.This shows that the OBEs lead to an Einstein coefficients master equation when the transverse

relaxation is large, even tough the driving field is nearly monochromatic (let us recall that the Einsteinargument a priori holds for a stochastic field with a large spectral bandwidth).


This is particularly reassuring for the laser theory that we briefly outlined. When a laser operates,the intra-cavity field is intense and nearly monochromatic. We would expect thus a regime of Rabioscillations. The Einstein coefficients provide a precise description of the laser operation, preciselybecause, in most laser media, the transverse relaxation is intense. In gaz lasers, the collisions dampefficiently the laser coherence. In solid-state lasers, inhomogeneous broadening causes a rapid relax-ation of the transverse components of the polarization, equivalent to a strong transverse relaxation.In simple words, something must have a large spectrum for the Einstein coefficients approach to bevalid, either the field as in the previous Chapter, or the atom, due to a strong transverse relaxation.

It should be noted that, in the case of some lasers, the transverse relaxation is smaller or muchsmaller than the Rabi oscillation. This is for instance the case in the micromaser, that will be brieflydiscussed in the last Chapter. The dynamics of these lasers is then radically different from that ofthe ordinary ones (in particular the notion of threshold must be radically modified), and it cannot bedescribed in terms of Einstein’s coefficients.

Stochastic fields

We now show how the OBEs lead to the Einstein’s coefficients in the case when the field has a largespectral width. We thus consider the case of a field with a stochastic amplitude and phase, whoseaverage frequency ω is nevertheless close to that of the atomic transition. Such a field can be writtenin terms of a slowly variable stochastic complex amplitude E1(t)9:

E(t) = E1(t)e−iωt . (2.155)

Such a field would, for instance, be quite appropriate for describing the radiation of a spectral lamp.This radiation is made of small pulses of light, corresponding to emission by individual atoms, all closeto the atomic transition frequency, with random phases between them. It is furthermore possible toassume that each of these tiny pulses has an exponentially decaying amplitude.

The properties of stochastic fields are described by the autocorrelation function ΓE(τ), defined as:

ΓE(τ) = limT→∞

1

T

∫ t+T

tE∗1(t′)E1(t′ − τ) dt′ , (2.156)

where the integral depends only upon τ in the limits of large T s for stationary stochastic processes,whose statistical properties are time invariant.

There is a useful alternative formulation of the autocorrelation function that holds provided wemake an ergodic hypothesis. Instead of taking the average of the fields product over a long time period,we can equivalently take an ensemble average at a given time over a large number of realizations ofthe source (a large ensemble of spectral lamps for instance). We thus assume that:

ΓE(τ) = E∗1(t)E1(t− τ) , (2.157)

where the overline symbol denotes the ensemble average. Note that:

ΓE(−τ) = E∗1(t)E1(t+ τ) = E∗1(t′ − τ)E1(t′) = Γ∗E(τ) . (2.158)

For zero time intervals, τ = 0, the autocorrelation function reduces to a term proportional to theintensity of the source. At long times, ΓE goes towards zero since the fields radiated at the two timesdo not have fixed phase relations. In between, ΓE(τ) is generally a decreasing function of τ , with acharacteristic width τc measuring the correlation time of the source.

9We of course consider here a single spatial point, where the atom is located. A full study of the spatio-temporalcorrelations properties is beyond the scope of this course. See the Born and Wolf Optics textbook for further information.


The spectral density of radiation, SE(ω), (the quantity of energy in a narrow band dω around thefrequency ω) is defined as the Fourier transform of the autocorrelation function10:

SE(ω) =1

2π

∫ ∞−∞

dτ ΓE(τ)e−iωτ , (2.159)

which is real due to the relation (2.158). The spectral density is centred around zero frequency, witha width of the order of 1/τc. The spectrum of the source is the spectral density translated in thefrequency space by ω.

Assuming that E1 varies slowly at the scale of the optical frequency (the field is close to beingmonochromatic), we can just insert this time-dependent amplitude in the OBEs for the density matrixelements and write:

dρegdt

= −i∆ρeg − γ′ρeg −id

2hE1(t)(ρee − ρgg) , (2.160)

where ∆ is now ωeg − ω.In order to get rid of the first two trivial terms in the above equation, we define:

ρeg = ρege(i∆+γ′)t . (2.161)

This is equivalent to an interaction representation (albeit with a complex evolution frequency). Wecan then formally integrate the evolution equation, with the initial condition ρeg(0) = 0 since weassume as usual that the atom is initially in |g〉. We then get:

ρeg(t) = − id2h

∫ t

0E1(t′)(ρee − ρgg)(t′)e(i∆+γ′)t′ dt′ . (2.162)

Returning to the ρeg = ρeg exp[−(i∆ + γ′)t] variable, we get

ρeg(t) = − id2h

∫ t

0E1(t′)(ρee − ρgg)(t′)e(−i∆−γ′)(t−t′) dt′ . (2.163)

We then plug this formal expression in the evolution equation of the populations and use Im i = Re :

dρeedt

= −Γρee −d2

2h2 Re

∫ t

0E∗1(t)E1(t′)(ρee − ρgg)(t′)e(−i∆−γ′)(t−t′) dt′ . (2.164)

Settting t− t′ = τ , or t′ = t− τ (0 ≤ τ ≤ t), this can be rewritten as

dρeedt

= −Γρee −d2

2h2 Re

∫ t

0E1(t− τ)E∗1(t)(ρee − ρgg)(t− τ)e(−i∆−γ′)τ dτ . (2.165)

We now perform an ensemble average of this equation over many realizations of the source. Ob-viously, the density matrix should be invariant in this operation, since its predictions are relative toa collection of very many identical systems. The averaging makes the field autocorrelation functionappear in the r.h.s. and we get:

dρeedt

= −Γρee −d2

2h2 Re

∫ t

0ΓE(τ)(ρee − ρgg)(t− τ)e(−i∆−γ′)τ dτ . (2.166)

The autocorrelation function has a width τc. We assume this time scale to be the shortest in theproblem (it is of course nevertheless long compared to the optical frequency). In the above integral,we can thus replace (ρee − ρgg)(t − τ) by (ρee − ρgg)(t). Moreover, since the times τ much greater

10Note that we use here an asymmetric definition of the Fourier transform with a 1/2π factor instead of the standard1/√

2π.. This is the most commonly used convention in this domain.


than τc do not contribute significantly to the integral, we can extend the upper bound to infinity. Wefinally get:

dρeedt

= −Γρee − C(ρee − ρgg) , (2.167)

where

C =d2

2h2 Re

∫ ∞0

ΓE(τ)e(−i∆−γ′)τ dτ . (2.168)

In order to relate C to the incoming field spectral density, we first neglect the transverse relaxation rateγ′ in this equation (we assume in the present regime that the dominant cause of spectral broadeningis the spectral width of the field, and we can thus safely neglect that of the atomic transition):

C =d2

2h2 Re

∫ ∞0

ΓE(τ)e−i∆τ dτ . (2.169)

The worrying point is that the integral only extends from zero to infinity. It is thus not ex-actly identical to that giving the spectral density of the field at the detuning frequency ∆. We cannevertheless write:

2πSE(∆) =

∫ 0

−∞ΓE(τ)e−i∆τ dτ +

∫ ∞0

ΓE(τ)e−i∆τ dτ . (2.170)

With a change of sign of the integration variable and using ΓE(−τ) = Γ∗E(τ):∫ 0

−∞ΓE(τ)e−i∆τ dτ =

∫ ∞0

ΓE(−τ)ei∆τ dτ =

(∫ ∞0

ΓE(τ)e−i∆τ dτ

)∗. (2.171)

Hence,

2πSE(∆) = 2Re

∫ ∞0

ΓE(τ)e−i∆τ dτ . (2.172)

and, finally

C =πd2

2h2SE(∆) , (2.173)

a very simple result indeed.The spectral density is obviously proportional to the spectral energy density uv at the atomic

frequency ω + ∆. Including the proper factors,

uν = 2π2ε0SE(∆) . (2.174)

We can thus writeC = Beguν , (2.175)

with

Beg =d2

4πε0h2 (2.176)

and recover for the population equations those of the Einstein model:

dNe

dt= −AegNe +Beguν(Ng −Ne) . (2.177)

The Beg value obtained here is somewhat different from that that can be deduced from the linkbetween the Aeg and the Beg Einstein’s coefficients and from the precise quantum value of the quantumspontaneous emission rate (see Chapter 4):

Aeg =d2ω3

3πε0hc3, (2.178)


leading to:

Beg =d2

6πε0h2 , (2.179)

a factor 2/3 off from the expression above. The difference is due in part to the fact that, here, we haveconsidered an incoming wave with a well defined polarization. In order to get the proper Einsteincoefficient, we should average the result over all possible polarizations (the Einstein approach dealswith a disordered radiation).

As an exercise on correlation functions, let us now compute that of the field emitted by a spectrallamp. We thus consider emissions by independent atoms. In a semi-classical approach, guided by thecharged oscillator model, we will assume that each atom emits at a random time a field pulse at theatomic frequency ω, with an exponentially damped amplitude and a random phase. The global fieldof the source is the sum of these independent pulses:

E1(t) =∞∑

i=−∞E0e

iφie−(t−ti)/τeΘ(t− ti) , (2.180)

where the index i enumerates all the individual pulses. The Heavyside function Θ ensures that eachpulse has a zero field before it starts. The real E0 is the common amplitude of all the field pulses,which have phases φi, start at time ti and are damped with an emission time constant τe (which isclearly a priori of the order of the source correlation time τc. We note Np the number of pulses perunit time.

The correlation function ΓE(τ) is given by the equation (2.156). Expanding the two fields atdifferent times, we find square terms describing the autocorrelation of an individual pulse and crossterms involving the product of fields generated in different pulses. Since the pulses have randomrelative phases, it is quite clear that the cross terms average out to zero and that only the squareterms contribute to the final result. The autocorrelation function is proportional to that of a singlepulse:

ΓE = NpTγe , (2.181)

with

γE(τ) =1

TE2

0

∫ T

0e−t/τee−(t−τ)/τeΘ(t− τ) dt . (2.182)

We will take the T →∞ limit at the end of the calculation. We set τ > 0 and will use the symmetryof the correlation function to get the values for τ < 0. We have assumed without loss of generalitythat the pulse starts at time t = 0. The delayed field is a damped exponential starting at time τ(hence the Θ(t− τ) function. During a time interval T , we have NpT contributing pulses and we thushave ΓE(τ) = NpTγE(τ).

The computation of γE(τ) is then a simple integration exercise. We get rid of the Θ function bychanging the lower integral bound from 0 to τ , extend the other bound to ∞ since T τe and get:

γE(τ) =1

TE2

0

[∫ ∞τ

e−2t/τe dt

]eτ/τe

=1

TE2

0

τe2e−|τ |/τe , (2.183)

where we introduce a |τ | to make the expression correct for negative τ values as well. Introducing thenormalization factor NpT and recognizing that the T →∞ limit is trivial, we get finally:

ΓE(τ) = NpE20

τe2e−|τ |/τe , (2.184)

which allows us to identify the emission time τe with the source’s correlation time τc. The correlationfunction has a bi-exponential shape that reflects the envelope of the individual pulses. A simple Fourier


transform leads to the spectral density:

SE(ω) =NpE

20

π

1

ω2 + (1/τe)2. (2.185)

We should remind ourselves that the actual spectrum of the source is this spectral density translatedby the average emission frequency ω. The source has thus a Lorentzian spectrum, whose width islimited by the pulse emission time, a rather direct consequence of the Fourier transform properties,or, in more shiny worlds, of the Heisenberg time-energy uncertainty relations. For a typical opticaltransition, the emission time is in the 10 ns range and the spectrum width is thus in the 16 MHzrange.

Note that we have assumed here that all atoms in the source emit pulses at the same frequency.This is far from being the case in most actual atomic sources. First, inhomogeneous emission lineshifts, due for instance to electric fields in discharge lamps, may contribute to a broadening of thespectrum. Most importantly, we have not taken into account here the Doppler effect due to atomicmotion if the emitting atoms are in a gas phase. It is of the order of a few hundred MHz for an opticaltransition and atoms in a vapour at room temperature.

Doppler effect could be taken into account in our calculations by adding a slow oscillatory compo-nent (at the typical Doppler shift frequency) in all pulses and by computing the resulting autocorrela-tion. Since the cross terms between different pulses do not contribute, we should get, for each velocityclass in the sample, the same correlation function as above, centred at the Doppler-shifted frequency.The total correlation function is thus the average of these shifted bi-exponential correlation functions,weighted by the velocity distribution.

The spectrum is thus a convolution of the Lorentzian emission-time limited shape by the Gaussiandistribution of Doppler shifted frequencies (a Voigt profile). There are two simple limiting cases.When 1/τe is much smaller than the Doppler width, the spectrum is a Gaussian. This is for instancethe case in the very low pressure spectral lamps, such as the historical Geissler tube. The other limitis when τe is very short. The spectrum is then Lorentzian, and its width dominated by the emissiontime broadening. This occurs in two limiting cases. The first occurs when the Doppler effect is verysmall, either because the atomic velocities are small (emission by an ensemble of laser cooled atom)or because one observes the fluorescence of an atomic beam at a right angle with respect to its meanvelocity. The second limiting case occurs when the coherence time of the source, τe is reduced, forinstance because of collisions between excited atoms or between excited atoms and a background gas.This is the case in most high-pressure gas lamps (sodium lamps for street lightning) in which thedischarge operates at very high pressures. The spectrum is then again mostly Lorentzian, with awidth in the GHz range or above.

2.6 Applications of the optical Bloch equations

We now focus on a few direct applications of the Optical Bloch equations. We first examine thesaturation of an optical transition by a strong field in the high-transverse relaxation regime, leadingto features absent from the classical model of the first Chapter. We describe optical pumping, whichallows one to act on the internal state of a multi-level atom through its interaction with a laserfield. We then turn to coherent excitation phenomena, dark resonance and eletromagnetically-inducedtransparency (EIT). They rely on subtle quantum interference phenomena and lead to interestingapplications such as slow light propagation and coherent transfer of quantum information betweenlight and atomic ensembles. We finally supplement the OBEs with the classical Maxwell equationsand write the full equations of propagation of a classical light in a quantum atomic medium.

2.6. APPLICATIONS OF THE OPTICAL BLOCH EQUATIONS 53

2.6.1 Steady-state and saturation

We consider first the simple problem of the energy exchange between matter and field. We have already,in the first Chapter, written the power delivered to the matter by the field, in the harmonically boundelectron model. We recall here the main result [see Eq.(1.30)]: the density of energy released by thefield in the matter is E = 1

2ε0ωχ′′|E1|2 = 1

2ε0ωNα′′|E1|2, where N is the number (or density) of atoms,

ω the incoming field frequency, E1 its complex amplitude and α′′ the imaginary part of the atomicpolarizability. We plan to compare this simple expression to the predictions of the OBEs.

Let us first turn the expression of E in a slightly different form. Noting that the complex dipoleamplitude is D = ε0αE1, we can write E as:

E = N ωE1

2ImD , (2.186)

where we assume in the following for the sake of simplicity, without real loss of generality, that E1

is real. The power released is proportional to the incoming intensity. Classical atoms can radiatearbitrary power levels. This should not be the case for a quantized two-level atom.

In the semi-classical model, we will replace in the above expression of E the dipole by D [Eq.(2.133)].Whatever the precise values of the relaxation coefficients are, the atom will reach after some time asteady state. The corresponding dipole value is given by Eq.(2.151):

D =∆ + iγ′

∆2 + γ′2d2E1

h(Ng −Ne) . (2.187)

In this regime, the steady state value of the upper state population11 is:

Ne =d2E2

1

2h2Γ(Ng −Ne)

γ′

∆2 + γ′2, (2.188)

and, from Ng +Ne = 1:

Ne =s/2

1 + s(2.189)

Ng −Ne =1

1 + s, (2.190)

where we define the ‘saturation parameter’, s by:

s =Ω2

Γγ′1

1 + ∆2/γ′2, (2.191)

and making finally use of the Rabi frequency Ω = dE1/h. Note that in this regime (with no externalpumping on the atoms), Ng − Ne is always positive as in the classical case. The medium is alwaysabsorptive. The saturation parameter varies, as a function of the detuning ∆ as a Lorentzian, whosewidth is given by γ′, a rather intuitive result. We get from there finally the average value of the dipoleD, whose square modulus |D|2 in the steady state is:

|D|2 = d2 Γ

γ′s

(1 + s)2. (2.192)

Using the expression of Ng −Ne and plugging that of D into E , we finally arrive at :

E =N hωΓ

2

s

1 + s, (2.193)

11We deal here with the OBE populations (which sum to one), not to be mistaken with the total number of atoms N


Clearly, s is large when the Rabi frequency dominates the evolution, small when the Rabi frequencyis small compared to the relaxation rates. For small values of Ω, the shape of the absorption curveversus ∆ follows that of the saturation parameter s and is thus also a Lorentzian. For high intensities,the width of the resonance curve increases and its top flattens, since E cannot exceed NhωΓ/2 (seebelow). At resonance (∆ = 0), we have:

s = s0 =Ω2

Γγ′. (2.194)

For low input powers,the power absorbed by the matter is proportional to s0, i.e. to the squareof the incoming field amplitude. We thus basically recover the results of the classical model. Forlarge s0 values, the power saturates to a finite value. This saturation in stark contradiction with thepredictions of the classical model. The limit for very large s0 is:

Es = N hωΓ

2. (2.195)

It has a simple interpretation. The power radiated by atom is one photon (hω) each time interval2/Γ. At saturation, the populations of the two levels are equal and, hence, both 0.5 on the average.An atom in |e〉 radiates one photon in a time 1/Γ. Hence a saturated atom radiates at a rate 2/Γ.

The transition between the classical linear regime and the saturation regime occurs for s0 ≈ 1. Letus discuss the physical signification of this s = 1 condition. Noting that s0 = d2E2

1/h2Γγ′ and that

the incident power per unit surface is I = ε0cE21/2, we can write

s0 =d2E2

1

h2Γγ′=

I

Is, (2.196)

where the saturation intensity Is is defined as

Is =Γγ′

d2

ε0c

2h2 . (2.197)

In order to estimate the order of magnitude of the saturation intensity, we will consider the caseγ′ = Γ/2 (no additional transverse damping):

Is =Γ2

4

ε0c

d2h2 . (2.198)

Let us plug in the exact quantum expression of the spontaneous emission rate (see Chapter 4) Γ =ω3d2/3πε0hc

3, After a bit of algebra, and noting that ω2/c2 = 4π2/λ2, where λ is the transitionwavelength, we get:

Is =π

3hωΓ

1

λ2= hω

Γ

2

1

σc, (2.199)

where we have introduced again the classical resonant absorption cross section σc = 3λ2/2π. Thesaturation intensity corresponds thus to a classical absorption on one photon at a rate Γ/2, themaximum power a quantum atom can diffuse

Taking a typical atomic transition, with Γ = 3. 107 s−1 i.e. an excited lifetime of 30 ns, a photonenergy of 1.5 eV, corresponding to a 1 µm wavelength (these orders of magnitude are relevant for theprincipal line of the Caesium atom), we get Is = 6 W/m2 or Is = 0.6 mW/cm2. This is a rather lowpower, easily reached with standard laser sources. Driving such a line far above saturation is easy. Itshows that the classical model is inappropriate in most cases, not a surprising result.

When the saturation parameter is large, the absorbed power (of the order of Is) is much lowerthan the input one. A strong drive makes an atomic medium nearly transparent. This leads to quiteinteresting applications, in particular saturated absorption spectroscopy.


Laser Cell

Detector

(a) (b)

Figure 2.2: (a) Typical saturated absorption set-up. (b) Saturated absorption spectrum on the transition between the

F = 1 ground state and the F ′ = 0, 1, 2 excited levels. The signal is proportional to the detected intensity.

Laser spectroscopy in a vapour has to face the problem of the Doppler broadening, a few hundredMHz at room temperature for a gas. This is much more than many interesting structures, such as thehyperfine splittings in excited states. This is of course also much larger than the spectral width, ofthe order of 1/Γ, due to the finite lifetime of the excited states.

Doppler-free spectroscopy is thus badly needed. There are a few techniques available. The mostancient ones rely on atomic beam technology, the interrogation laser propagating then at a rightangle with the beam. Only the second-order relativistic Doppler effect is left, much less important inmagnitude (being of order v2/c2 ≈ 10−12 for thermal velocities). Laser cooling can also be used, butthese two methods imply rather large and complex apparatus. For simple experiments, compatiblewith atoms contained in a small cell at room temperature, Doppler-free two photon and saturatedabsorption are to be preferred. The second is most widely used, since it can be applied to almost anyatomic transition.

The typical set-up for saturated absorption is shown on figure 2.2(a). A cell contains the resonantatoms. The laser beam at frequency ω propagates through the cell, is reflected back onto a mirror andthe returning beam is separated from the incoming one by a beam-splitter (in most cases, one usespolarizing beam-splitter and wave-plates to separate the incoming and outgoing lases without signallosses). The final power of the outgoing laser is measured with a quadratic detector.

It is important to realize that, due to the Doppler effect, only one velocity class is exactly resonantwith the incoming laser. Calling z the propagation axis (oriented in the direction of the incominglaser), the resonance condition is obtained when ∆ = ωeg−ω = −kvz, where k is the laser wave-vectorand vz the z component of the atomic velocity.

In a first qualitative picture, all velocity classes are impervious to the incoming laser, but the onethat is on resonance. A part of the incoming laser is absorbed. Close to saturation, it is proportionalto the density of atoms in the velocity class. Measuring the single pass absorption clearly leads to theDoppler-broadened spectrum.

Let us now consider the return beam. It is also resonant with a single velocity class, with avelocity v′z given by the resonance condition ∆ = kv′z (note that this laser propagates in an oppositedirection). If the resonant classes for the two lasers are different, the absorption on the return pathadds to the absorption on the incoming path. However, when the two paths address the same velocityclass, the absorption of the return beam is reduced due to the saturation induced by the first. Becauseof saturation, the absorptions do no simply add as in the linear classical model. Hence, one should


observe a narrow reduction of the overall absorption when the two beams address the same velocityclass, i.e. when vz = 0, and hence ∆ = 0. This saturated absorption transmission peak is centredprecisely on the atomic frequency and appears on the wide background of the Doppler-broadenedabsorption [figure 2.2(b)].

More quantitatively, out of resonance (∆ much larger than Ω and Γ), the two counterpropagatingbeams interact with different velocity classes due to the Doppler effect. The absorptions are inde-pendent and equivalent to one path in a medium with a double density 2N . The absorbed energyis

E = 2× N hΩΓ

2

s0

1 + s0(2.200)

At resonance (∆ = 0), the two beams interact with the vz = 0 class. The saturation parameter isdoubled (twice the intensity) but the density is twice lower (only one class). The absorbed energy isthen:

E0 =N hΩΓ

2

2s0

1 + 2s0(2.201)

Hence, the ‘dip depth’ isE0

E=

1 + s0

1 + 2s0(2.202)

and its width is γ′√

1 + s0.To observe the peak, one should be in a moderate saturation regime. For too low s0 parameters,

the absorptions add linearly, even when the two paths address the same velocity class. For too highs0 parameters, the medium is basically transparent and no absorption can be detected. The bestcompromise corresponds to s0 ≈ 1, with a dip depth of 1/3 and a width of 2γ′).

This simple picture does not give us the frequency resolution of the method. By consideringequation (2.193), we see that the width of the absorption line in the absence of Doppler effect isγ′√

1 + s0. This is also of course the order of magnitude of the saturated absorption transmission peak.For s of the order of unity, the width of the peak is thus set by the total transverse relaxation rateγ′, equal to Γ/2 when no other phenomenon than spontaneous emission contributes to the linewidth.The saturated absorption method thus reaches nearly the limit set by the excited level lifetime.

Saturated absorption spectroscopy has been widely used for high-resolution atomic and molecularspectroscopy. It is particularly interesting for infrared molecular lines, which can have extremely smallintrinsic linewidths. Saturated spectroscopy is also often used to lock lasers to atomic lines for long-term stability. The simple arrangement (one cell, one mirror, one beamsplitter and a detector) makesit very appealing for extensive use. A modern cold atom experiment optical table may contain up toten lasers, all locked to saturated absorption lines of an alkali atom.

In most cases, the atoms have more than two levels. This is particularly the case for most alkalis,who have a rich hyperfine structure. In this case the saturated spectrum may exhibit a more complexstructure, due to the presence of crossover resonances.

In order to understand them, we qualitatively consider a simple three level model. Our atom hastwo nearly degenerate ground states, |g〉 and |f〉, and an excited state |e〉 coupled to both groundlevels by dipole transitions at the respective frequencies ωge and ωfe.

The saturated absorption spectrum will obviously present transmission peaks at both ωge and ωfe.There is another additional feature. Let us consider the velocity class resonant on the ωfe frequencyfor the first pass in the cell. Its velocity is given by the resonance condition kvz = ω − ωfe. Thelaser being at saturation, it promotes a large fraction of the atoms initially in |f〉 into the excitedlevel |e〉. Hence, for this velocity class, the absorption of a laser resonant on the |g〉 → |e〉 transitionwill be reduced. The return path is resonant on this latter transition and this velocity class providedkvz = −(ω − ωge). Hence, we get a ‘cross-over’ transmission peak if ω − ωfe = −(ω − ωge) i.e. if

ω =ωfe + ωge

2, (2.203)


at the middle between the two standard saturated absorption features. There are three lines in thespectrum.

2.6.2 Optical pumping

Optical pumping covers a rather large variety of techniques whose aim is to use light to control theinternal state of the atoms. For instance, it is used to prepare the atoms in a single magnetic quantumnumber state, i.e. to control the internal angular momentum of the atoms. It has been originallyproposed by A. Kastler in 1950. It has been observed for the first time in 1952 by J. Brossel on asodium atomic beam, and in a cell by B. Cagnac in 1961. It led since then to considerable experimentaldevelopments. Optical pumping is still an important tool for laser manipulation of atoms and playsa key role in atomic cooling, whose aim is to control also the external degrees of motion of the atom,i.e. its motion.

The general theory of optical pumping occupies entire books. We will limit ourselves to a simplemodel, with a three-level atom as above. The excited level |e〉 may decay by spontaneous emissiontowards both the ground state levels |g〉 and |f〉 with respective rates Γeg and Γef . We start from asituation in which both ground levels are populated and want to reach one in which only level |f〉 ispopulated.

The simple idea of optical pumping is to drive selectively with a light source (a laser or, in theearly days of optical pumping, a spectral lamp) the |g〉 → |e〉 transition. From |e〉 either the atom fallsback into |g〉 by spontaneous emission, and nothing has been done, or it falls back to |f〉 and staysthere since the excitation source is selective. It is clear that, after a few absorption/emission cycles,the atom will eventually end up in |f〉.

The excitation selectivity on the |g〉 → |e〉 transition can be achieved through the source frequency,if the difference between ωge and ωfe is large enough. This is for instance the case when pumping inone of the hyperfine ground state sublevels of an alkali atom. The typical hyperfine splitting being ofa few GHz, the structure can be resolved with a laser or even with a spectral lamp. When the twofrequencies are too close, one can use instead the polarization selectivity of the atomic transitions.This is the case for the optical pumping of the magnetic sublevels. Then, the polarization of the sourceis chosen to couple to a single transition.

Let us make this simple model more quantitative with an Einstein coefficient approach (strongtransverse relaxation). We need to generalize equation (2.153) in the large γ′ limit for this threelevel situation. It is not too difficult to guess the right form of the rate equations for the three levelspopulations:

dNe

dt= −(Γeg + Γef )Ne +

Ω2

2γ′(Ng −Ne) (2.204)

dNg

dt= ΓegNe −

Ω2

2γ′(Ng −Ne) (2.205)

dNf

dt= ΓefNe , (2.206)

with 1 = Ne +Nf +Ng, where Ω is the Rabi frequency on the driven transition.The steady state can be obtained easily. From the last equation, we get Ne = 0. Plugging in the

second, it leads to Ng = 0 and, hence, to Nf = N . In the steady state, all the atoms are in level |f〉as intuitively expected.

Let us now examine the dynamics towards the steady state. We take as initial condition Ng = N ,without loss of generality (if some atoms are already in |f〉, they will not take part in the dynamics).We also assume, for the sake of simplicity, a weak pumping limit:

Ω2

γ′ Γeg, Γef . (2.207)


The population in level |e〉 is thus low. We can treat it as being at any time in a steady-state givenby the first equation:

Ne =Ω2/2γ′

Γeg + Γef + Ω2/2γ′Ng ≈

Ω2/2γ′

Γeg + ΓefNg . (2.208)

Plugging this result in the equation of evolution for Ng, we get:

dNg

dt= −ΓpNg , (2.209)

where we define the optical pumping rate Γp by:

Γp =Γef

Γeg + Γef

Ω2

2γ′. (2.210)

The population of level |g〉 decreases exponentially with a rate Γp, proportional to the sourceintensity, as expected in this rate equation model.

2.6.3 Dark resonances and EIT

In the previous paragraphs, we have considered multi-level systems in the simple case when theinteraction with light can be described in terms of rate equations. We now scrutinize in more detailsthis situation in the coherent evolution case. We will show that new and interesting phenomena arise,due to a quantum interference process between the competing atomic transitions. We will discuss theimportant phenomenon of electromagnetically induced transparency. We will also finally outline thestudy the atomic levels modifications by a resonant (Autler Townes effect) or non-resonant light field(light shift, which play an important role in laser spectroscopy).

Dark states or coherent population trapping

We consider again the three-level structure described above. We assume that two input fields atfrequencies ω1 and ω2, E1 and E2, couple separately (due to selection rules) to the two |g〉 → |e〉 and|f〉 → |e〉 transitions. The total interaction Hamiltonian is Hi = −d · (E1 + E2), where d is the dipoleoperator (back to the basics). We set

deg = 〈e|d ·E1|g〉; def = 〈e|d ·E2|f〉 (2.211)

We now show that one can define a superposition |Ψ−〉 of |g〉 and |f〉 that is not coupled to thefield. Let |Ψ−〉 = cg |g〉+ cf |f〉. It decouples from the field if 〈e|Hi |Ψ−〉 = 0 or cgdeg + cfdef = 0 i.e.

cg =defd

and cf = −degd

(2.212)

or

|Ψ−〉 =defd|g〉 − deg

d|f〉 (2.213)

withd =

√|deg|2 + |def |2 . (2.214)

The decoupling of the ‘dark’ state |Ψ−〉 with the field is clearly due to a quantum interferencephenomenon: excitation can occur either through the |g〉 component or through the |f〉 componentand these two competing excitation paths have amplitudes, proportional to the Rabi frequencies,which interfere destructively, leading to a vanishing probability for making the transition to |e〉.

Obviously the orthogonal state (in the |g〉 , |f〉 subspace):

|Ψ+〉 =d∗egd|g〉+

d∗efd|f〉 , (2.215)


is coupled to the driving field.If we switch to the |Ψ+〉 , |Ψ−〉 basis in the ground state manifold, we are left with one level

coupled to the excited state and another uncoupled. We are thus back very much in the opticalpumping situation. If we now include the relaxation from the excited level into both ground statesones and switch on the driving field, the atom will be rapidly pumped into the dark state which isdecoupled to the field. The fluorescence emission by the atom will stop and it will remain imperviousto the incoming light.

We have here written the states and the fields at a given time. The dark state condition will beindependent of time if the two lower levels are exactly degenerate and it the lasers are both at exactresonance. When |g〉 and |f〉 have different energies, or Bohr frequencies, ωg and ωf , the dark statesuperposition evolves in time as well as the electric fields. The dark state condition will be maintainedfor every time provided:

|Ψ−〉 (t) =defde−iω1te−iωgt |g〉 − deg

de−iω2te−iωf t |f〉 (2.216)

is, within a global phase, independent of time i.e; if ω1 + ωg = ω2 + ωf or

ω1 − ω2 = ωfg , (2.217)

if the difference of the fields frequencies is equal to the Bohr frequency between the two ground states.This is nothing but a Raman resonance condition.

The first evidence of dark resonances has been obtained in 1976 by Gozzini and his group in Pisa.They illuminated by two laser modes an atomic vapor placed in a gradient of magnetic field sweepingthe transition frequency with the position, and observed a narrow dark spot in the fluorescence inducedby the lasers where the Zeeman level splitting exactly matched the frequency difference of the twomodes.

Note that the dark resonances are closely related to the STIRAP technique, which allows a fastand efficient transfer from |g〉 to |f〉 by modulating with time the relative amplitudes of the two lasers.Using a counter intuitive sequence in which the laser coupled to |f〉 is switched off while the other hasa raising intensity, one can achieve the transfer by remaining at any time in the Dark state, avoidingthus any noise due to spontaneous emission from |e〉.

Electromagnetically induced transparency

We now examine another important quantum interference situation, where the interference is nowthe competition between two excitation paths driven by two classical fields. We thus return the thenon-degenerate case (ωge 6= ωfe) and assume that we drive the |f〉 → |e〉 transition with a strongpump field E at frequency ω (resulting Rabi frequency Ω) and simultaneously the |g〉 → |e〉 transitionwith a weaker proble field E′ at frequency ω′, and a Rabi frequency Ω′. We assume here that thelaser on the |f〉 → |e〉 transition is resonant, while the other may be detuned from exact resonance by∆′ = ωeg − ω′.

The OBE equations, including the spontaneous emission relaxation, cannot be obtained in a com-pletely trivial way. We would have to return to the basic Hamiltonian and Liouvillian ant to redo allthe initial derivations of this Chapter from the very start. We leave this lengthy path to the readeras an exercise and only give here the final result.

With self-explanatory notations, the equations of motion of the atomic density matrix relevantelements read:

dρeedt

= −(Γef + Γeg)ρee − ΩIm ρef − Ω′Im ρeg (2.218)

dρffdt

= Γefρee + ΩIm ρef (2.219)


Figure 2.3: EIT signals. Absorption of the probe field as a function of the detuning ∆ for Ω = 10Γ (left) or Ω = 0.1Γ

(right). The unit of the horizontal axis is Γ.

dρggdt

= Γegρee + Ω′Im ρeg (2.220)

dρefdt

= −Γef2ρef + i

Ω

2(ρee − ρff ) + i

Ω′

2ρgf (2.221)

dρegdt

= −(

Γeg2

+ i∆′)ρeg + i

Ω′

2(ρgg − ρee) + i

Ω

2ρfg (2.222)

dρgfdt

= −i∆′ρgf + iΩ

2ρge − i

Ω′

2ρef (2.223)

It is clear that the structure of these equations cannot be guessed easily from that of the standardtwo-level OBEs; In particular the coupling of the coherences to the populations contain non trivialterms, and the coherence between the two ground levels plays an important role.

Simplifying a bit these equations by assuming Γef = Γeg = Γ, it is possible to extract the absorptionof the probe field E′ (proportional to Im ρeg in the steady state. The absorbed energy E ′ is found tobe proportional to:

E ′ ∝ Γ2∆′2

Γ2∆′2 + 4(∆′2 − Ω2/4

)2 (2.224)

This rather complex expression is plotted in figure 2.3 in two simple limiting cases.When the pump field Rabi frequency Ω is large compared to the excited level width, we get two

lines separated by the Rabi frequency. This is the Autler-Townes splitting whose physical significancewill be discussed in more details in the next paragraph. When Ω is very small as compared to Γ, wefind an absorption profile with a width Γ (as for a two-level atom) with a dip in the centre, goingall the way to zero absorption. The width of this dip is of the order of Ω, and can be made nearlyarbitrarily small by reducing the pump laser power (not all the way to zero, since the approximationsperformed to derive the above expression of the absorption would not apply any longer for very lowΩs).

At exact resonance of the probe field, the pump field create a near zero absorption gap, hence thename Electromagnetically Induced Transparency coined for the phenomenon. It results from a subtlequantum interference between the absorption processes of the two lasers.

In these conditions, a narrow hole is created in the absorption profile. This makes EIT a usefulspectroscopic tool on the one hand. On the other hand, since absorption is proportional to the


imaginary part of the susceptibility χ, it has a fast variation in a narrow frequency range aroundresonance. In fact, the real part of χ has also a rapid variation in the transparency gap, which caneither be computed directly from the OBEs or through the Kramers Kronig relations relating the realand imaginary parts of the susceptibility.

A medium made of atoms in the EIT configuration has thus a refraction index for the probe beam,n(ω′), which varies extremely rapidly with the frequency. In particular, at exact resonance, n is arapidly growing function of ω′. Reminding ourselves that the group velocity, vg, for light propagatingin a medium is:

vg =c

n+ ω′ dn/dω′, (2.225)

we recognize that, close to resonance, the velocity of light is considerably reduced with respect to c.This ‘slow light’ phenomenon has been observed in a variety of media, such as transparent solids

doped with optically active ions, and in particular in cold atomic samples or Bose Einstein condensates.Considerable velocity reductions can be obtained, down to the m/s range. It is even possible, byswitching off the pump laser intensity while the light is inside the medium, to stop the light altogether.It restarts again at low speed when the pump intensity is restored.

Note that the conditions for the observation of the slow light effect are demanding. The whole lightpulse should fit in the medium (with an extension of a few hundred microns for BECs). It should thusbe short enough in time. But the pulse spectrum should also be all contained in the EIT transparencywindow and inside the high index variation region, which is as narrow as Ω, much lower than theoptical frequency. A careful optimization of the parameters is thus mandatory.

What does a stopped light (or such slow light) really means? In fact, the thing that propagates isnot pure light, but a complex mixture of light together with atomic spin coherence. In the stoppedlight case, the wave is stored in the medium as a spin wave, a coherent spatial arrangement of atomiccoherences.

Slow and stopped light can be used for the storage of quantum information carried by light intothe atomic medium for later retrieval. This is an important step for quantum key distribution orquantum teleportation operations for instance. These memories can now store individual photons andretrieve them later with a reasonable fidelity.

Of course the storage time is limited. In the cold atom case one of the main limitations, leadingto a storage time in the ms range, is the residual atomic motion. This motion blurs the spin patternwhich store the incoming light and the retrieval becomes impossible. Of course this effect does notexist in doped crystals, but other sources of inhomogeneous broadening also contribute to limiting thememory lifetime.

Light shifts and Autler Townes splitting

We turn for this paragraph only to a ladder configuration, with |g〉 as the ground state, |f〉 being moreexcited, and |e〉 even more so. We strongly drive the |f〉 → |e〉 transition at or close to resonance andwe probe with a weak drive the |g〉 → |f〉 one at ωgf . When the strong drive is off-resonance, thereis still a single resonance close to ωgf , but it is shifted with respect to the bare atomic frequency, byan amount proportional to the strong drive intensity. This is the light-shift effect, well known in laserspectroscopy.

When the strong drive is at resonance, the |g〉 → |f〉 transition is split in two lines, separated bythe Rabi frequency Ω of the strong drive, as in the EIT situation for strong drive. This Autler-Townessplitting can be understood in qualitative terms. The strong drive results in a fast oscillation on the|f〉 → |e〉 transition at frequency Ω. The population in |f〉 is thus modulated at Ω and, hence, theabsorption from |g〉 also modulated. This modulation creates sideband o the absorption, which areprecisely the Autler-Townes lines.

This discussion is obviously utterly qualitative. It can be studied in more details by the OBEsusing the formalism of the last paragraph,mutatis mutandis. However, the computations are complex


and we will see in Chapter 4 that the dressed atom picture provides a much more detailed and moreeconomic insight into these problems, by coupling a quantum atom to a quantum field. We will thusdefer their detailed study to the end of these lectures.

2.6.4 Maxwell-Bloch equations

We have up to now considered the driving field as imposed externally on the system. We now takeinto account the atomic medium response itself and write the equations for the propagation of lightin a medium of two-level atom whose density matrix obeys the OBEs.

We start by recalling a few results about propagation in dielectric matter. The Maxell equationsread:

∇×E = −∂B

∂t(2.226)

∇ ·D = 0 (2.227)

∇ ·B = 0 (2.228)

∇×B = µ0ε0∂E

∂t+ µ0

∂P

∂t, (2.229)

where E and B are the electric and magnetic field, P the density of polarization in the medium and Dthe electric displacement D = ε0E + P. The last term in the last equation corresponds to the boundcharges current.

We study the propagation of a transverse wave. All the fields are then divergence-free. By takingthe rotational of the first equation an using a standard vector calculs formula, we arrive at the equationof propagation of E:

∆E− 1

c2

∂2E

∂t2= µ0

∂2P

∂t2. (2.230)

This equation determines the propagation and the dispersion relation if P can be expressed as afunction of E. The standard approach is to use the linear response theory and to introduce the electricsusceptibility of the medium. We will use the OBEs instead to compute the polarizability.

Before doing that, we will simplify a bit this equation before proceeding further. We will consideronly a nearly monochromatic plane wave at frequency ω, propagating, without loss of generality, alongthe z axis. Assuming that the medium is isotropic and, thus, that P and E are collinear, we can writethem as:

P = P0(z, t)ei(kz−ωt)ux and E = E0(z, t)ei(kz−ωt)ux , (2.231)

where P0 and E0 are slowly varying envelopes (their variation describes slow phase or amplitudechanges of the wave due to the propagation). We set in these equations k = ω/c as for free spacepropagation (the phase shifts due to the index of refraction will be encapsulated in the slowly varyingenvelope).

We plug these expressions in the wave equation. The second derivatives of the field in z and tcancel due to our choice of dispersion relation. Noting that:

∂P0

∂t ωP0 , (2.232)

we can neglect the time derivatives of P0 in the r.h.s.. We get then, for the slow envelopes:

∂E0

∂z+

1

c

∂E0

∂t= i

µ0ω2

2kP0 . (2.233)

We can compute the local atomic density matrix as a function of E, or more simply as a function ofthe local field amplitude E0 by solving the OBEs. From that, we compute the density of polarization.


It is quite clear that, provided the density is large enough (many atoms contribute), it is proportionalto D and thus to the coherence of the atomic density matrix:

P0 = ND = 2Ndρeg , (2.234)

where N is the numerical density and where we assume again d real. We thus arrive at the ‘Maxwell-Bloch equations’, combining the OBEs with the propagation relation:

∂E0

∂z+

1

c

∂E0

∂t= i

ωNd

ε0cρeg . (2.235)

These coupled equations are rather complex and must generally be solved numerically. There aresome situations in which they can be simplified. We will describe one of them as an exercise.

We consider the propagation of a laser pulse in a medium of atoms, where we can neglect allrelaxation phenomena. The OBEs then describe only the Rabi oscillation of the atom between itstwo levels. Note that this approximation, very rough in most cases, is appropriate, for instance,to describe the propagation of a very short laser pulse (whose duration is much shorter than thespontaneous emission lifetime). The atoms are assumed to be all initially in their ground state |g〉.

In this simple situation, the atomic state is described entirely by the angle φ(z, t) of the Blochvector with the vertical axis (the axis of the Bloch sphere should not be confused with those of thereal space in which the wave propagates). This angle rotates at the local Rabi frequency and thus:

dφ(z, t)

dt=dE0(z, t)

h= Ω(z, t) . (2.236)

In order to avoid any difficulty, we shall define in z and at time t the phase of the atomic levels suchthat the Rabi frequency defined here, and accordingly the angle φ are real. This is always possible,since E0 varies slowly. We then have

ρeg = −i sinφ

2. (2.237)

Plugging this in the propagation equation, we can write the Maxell-Bloch equations as a simpleequation for φ:

∂2φ

∂z∂t+

1

c

∂2φ

∂t2= −µ sinφ , (2.238)

where

µ =ωNd2

2ε0hc. (2.239)

We can even further simplify this equation by using as independent variables z and the ‘retarded time’τ = t− z/c:

∂2φ

∂z∂τ− µ sinφ , (2.240)

This is the Sine-Gordon equation, a well-known non-linear propagation equation.The propagation in an atomic medium is thus a highly non-linear process. It is only the the

limit of a very low intensity, a very small angle of the Bloch vector, that we would recover an ordinaryindex of refraction. This non-linearity has many interesting properties. In particular, the Sine-Gordonequation sustains solitonic solutions.

Chapitre 3

Field quantization

3.1 Introduction

In this essential Chapter, we give a quantum nature to the field, before treating in the next andfinal Chapter the interaction of a quantum atom and a quantum field, completing our programme.We start in this short introductory section by recalling the original Planck and Einstein argumentsused to quantize the field energy and to built a consistent theory of the blackbody radiation. Wethen perform the field quantization itself. The first stage in field quantization is in fact a classicalelectromagnetic problem. We need to define independent, canonical variables for the field dynamics.We will thus devote a long section to the decomposition of the electromagnetic field in orthogonalmodes. We will write all the essential quantities, the field energy, the field momentum and the fieldangular momentum in terms of the dynamical variables of these modes.

We will then be ready to proceed to the field quantization. The analogy between the dynamics ofone mode and a one-dimensional harmonic oscillator will help us greatly, by making this quantizationa nearly trivial process. We will define the field operators, and examine in detail some interesting fieldstates. We will also define pictorial representations of the quantum field states in the phase space.

In the last two sections, we will supplement the elementary field dynamics, first by coupling twomodes with a simple beam-splitter. We will be able to unveil some interesting and deeply quantumphenomena such as the Hong Ou Mandel quantum interference process. The last Section of thisChapter will be devoted to field relaxation. We will guess the quantum jump operators for the field,write the Lindblad master equation and derive a few important consequences.

3.1.1 Planck 1900

The first apparition of the concept of field quantization dates from a communication by Planck atthe Berlin Academy of sciences, in December 1900. The interested reader should consult an historyof physics treaty to fully understand the problem, Planck’s position and the impact of his paper. Wewill only here give the standard vulgate of the physics textbooks.

Planck’s aim was to compute the spectrum of the blackbody field. A blackbody is fully absorbingat any wavelength. When heated to a non-zero temperature T , it emits radiation, as we all canexperience. Of course, the blackbody is an idealization. There is no such material that can be fullyabsorbing for all frequencies (be it only due to the Kramers Kronig relations). However, a goodapproximation of the blackbody radiation can be obtained by taking that emitted by a small holepierced in a closed oven, with black diffusive coating on its inner walls. In other words, calculatingthe blackbody spectrum amounts to computing the thermal equilibrium of the field at a temperatureT inside a large closed envelope.

This problem is not only of academic nature. It has also, and already had in Planck’s time,considerable industrial applications. Measuring the blackbody radiation is nearly the only way to

65

66 CHAPITRE 3. FIELD QUANTIZATION

get information on the temperature of a furnace or of a melting metal. A great deal was knownexperimentally at Planck’s time. The spectrum had been carefully measured and calibrated in thevisible and in the near-infrared. The radiation spectral density, uν (see Chapter 1) starts as ν2 for smallfrequencies, reaches a maximum and finally decays exponentially at high frequencies. The maximumof the intensity is in the 10 µm wavelength region for room temperature. Its frequency is proportionalto the temperature. It is thus in the visible for the solar chromosphere temperature (6000 K).

This density of radiation is universal. It does not depend upon the detailed nature of the envelopecontaining the radiation. A simple classical thermodynamics argument can be used to show it. Assumethat, for at least one frequency ν0, we could build two ovens such that they do not have the samedensity of radiation at the same temperature T . Joining the two ovens by a small channel (connectedthrough filters allowing only the frequency ν0 to flow freely), we would establish a constant flow ofenergy from one oven to the other. This flow can produce work, and we have thus built an engineworking with a single temperature source, an impossibility due to the second principle.

The total power radiated by a surface S was also well known to be proportional to the fourthpower of the temperature. This is the Stefan’s law:

P = σST 4 , (3.1)

whereσ = 5.67 10−8 W/m2K4 . (3.2)

As an order of magnitude, one square centimetre of blackbody at room temperature radiates about56 mW, a quite appreciable amount. The radiation of the blackbody obeys Lambert’s law. The powerdP radiated in the solid angle dΩ making the angle θ with the normal to the surface is:

dP = LS cos θ dΩ , (3.3)

where the luminance L is related to the total density of energy in the oven, u =∫uν dν, by:

L =cu

4π(3.4)

By a simple integration, we thus get

P =cSu

4, (3.5)

and hence

u =4

cσT 4 . (3.6)

On the theoretical side, little was known, even though statistical physics and electromagnetism werealready quite mature at this time. The difficulty is that classical statistical physics fails to computethe equilibrium state of the field. In a nutshell (more on that in a second), we should attribute anenergy kbT to each degree of freedom, i.e. to each field mode (to each plane wave, or to each standingwave pattern). But there is an infinity of modes and their number raises with the frequency. Classicalstatistical physics thus predicts boldly that the energy density is infinite, an embarrassing conclusion.

Using careful considerations on the entropy of the radiation (which will be extremely useful forEinstein in his 1905 paper, see next paragraph), Wien had show his ‘displacement law’, a scaling lawfor the spectral density:

uν = ν3f

(ν

T

), (3.7)

where f is a function that Wien could not explicit. He also proposed a semi-phenomenological modelfor high frequencies:

uv = αν2e−γν/T , (3.8)

where α and γ could be fitted to the exponentially decreasing part of the spectrum at high frequencies.

3.1. INTRODUCTION 67

Let us reproduce (more or less) Planck’s computation of the statistical equilibrium of the field.As will be explained in much more details in the following Section, the field should be expanded onmodes, orthogonal harmonic solutions of the Maxwell equations with boundary conditions, such thatthe total field energy is the sum of the energy of the modes.

For a simple approach, we will here assume that the oven is rectangular, with lengths Lx, Ly andLz along the three axes and a volume V = LxLyLz. We use periodic boundary conditions on the wallsof the oven. The field can only be expanded on plane waves whose wavevector k = (kx, ky, kz) is suchthat

kx =2π

Lxnx , (3.9)

and similar relations for the other components, where nx,y,z is a set of positive or negative integers.Each set of nx,y,z values is associated to two modes, two plane waves with orthogonal polarizations.

This set of plane waves is a basis for all solutions of the Maxwell equations. Moreover, since thescalar product of modes with different nx,y,z values (volume integral of the product of one mode bythe complex conjugate of the other) is zero, the energy of the field is the sum of the energies stored inall the modes (this derivation will be performed in great details in the next Section).

We note Nν the total number of modes with frequency between 0 and ν, i.e. with a wavevectork < 2πν/c. We introduce the spectral density of modes ρν . The number of modes per unit volumebetween ν and ν + dν is then ρν dν. Hence,

ρν =1

VdNν

dν. (3.10)

The total number of modes Nν is simply, within the factor of 2 due to the polarization, the number ofpoints with integer coordinates inside a sphere with a radius 2πν/c. The unit cell of this cubic latticeof points has a volume 8π3/V. It contains 8 vertices, but each vertex is shared by 8 adjacent cells.The volume occupied by a point is thus also 8π3/V. We thus have:

Nν = 2

4π3

(2πνc

)2

8π3

V=

8π

3

ν3

c3V , (3.11)

and hence

ρν =8π

c3ν2 . (3.12)

The spectral density of modes is proportional to ν2.If we apply then the standard statistical physics approach, we will attribute an average energy kbT

to each mode. The spectral density of radiation is then:

uν = kbTρν . (3.13)

This is the Rayleigh-Jeans law. The onset at low frequency fits rather well with the observed spectrum,but the high-frequency part is totally irrelevant, since uν diverges as already explained.

In order to circumvent this difficulty, Planck uses a state counting ansatz. He assumes that thematter-field energy exchanges in the oven are quantized and only proceed at frequency ν in integermultiples of a quantity, the light quantum hν. The field energy at frequency ν is then E = nhν, wherethe ‘hilfe’constant h has the dimension of an action and n is the integer number of light quanta. It is asimple exercise in statistical physics to compute the average number n of these elementary excitationsand hence the spectral density.

The average energy per mode E is given by the Boltzmann distribution:

E = hν

∑∞n=0 ne

−nhν/kbT∑∞n=0 e

−nhν/kbT. (3.14)


Setting the shorthand notations β = 1/kbT and χ = βhν, we note that∑

exp(−χn) = 1/[1−exp(−χ)]and

∑n exp(−χn) = −(d/dχ)1/[1− exp(−χ)] = exp(−χ)/[1− exp(−χ)]2. Hence

E = hνn = hν1

eχ − 1. (3.15)

Inserting this in the expression of the spectral density, we finally get the Planck’s law:

uν =8πhν3

c3

1

ehν/kbT − 1, (3.16)

whose form agrees with the Wien’s displacement law. Setting

h = 6.62 10−34 J/s , (3.17)

Planck obtained a perfect agreement with the available experimental spectra, a considerable break-through.

For small frequencies, the Planck law reduces to

uν =8πν2

c3kbT , (3.18)

the Rayleigh-Jeans law. For low frequencies, the energy of the light quantum (let us call it anachro-nistically a photon) is small as compared to the thermal energy and the quantization ansatz of Planckplays no role. We recover the standard statistical physics result.

At large frequencies, the spectral density decreases exponentially. It writes

uν =8πhν3

c3e−hν/kbT , (3.19)

in agreement with the phenomenological Wien’s law.

From the spectral density, we can compute the total power radiated by the unit surface of theblackbody. The integration is tedious (use Mathematica!), but gives the theoretical value of theStefan’s constant:

σ =2π5

15

k4b

c2h3, (3.20)

whose numerical value agrees remarkably with the experimental one.

Planck thought his quantization ansatz was only a way to compute in a proper way the spectraldensity. He did not take seriously the fact that the field energy itself could indeed be quantized,that the field might exhibit a corpuscular nature. This point was convincingly made 5 years later byEinstein.

3.1.2 Einstein 1905

In one of the three famous 1905 papers, Einstein tackles the problem of the thermal equilibrium ofthe field. He starts equipped with the Wien’s phenomenological law, in which, according to Planck,γ = h/kb:

uν = αν3e−hν/kbT = αν3e−γνT , (3.21)

with γ = h/kb. This leads by a simple inversion to:

T = − γν

lnuν/αν3. (3.22)

3.2. FIELD EIGENMODES 69

He introduces then the density of entropy of the field s. Since, for all systems, 1/T = ∂S/∂U ,ds/du = 1/T . Replacing T by the expression deduced from the Wien’s law and performing theintegration, he finds:

s = −∫ ∞

0du′

lnu′/αν3

γν

= − u

γν

[ln

u

αν3− 1

]. (3.23)

The total entropy S = sV (including only the contribution from the radiation at frequency ν) is linkedthus to the total energy E = uV by:

S = − Eγν

[ln

E

Vαν3− 1

]. (3.24)

Einstein now considers the variation of the entropy with the total volume V. Calling S0 the entropyfor the volume V0, he writes:

S − S0 =E

γνlnVV0

. (3.25)

This compares directly to the variation of the entropy of a perfect gas in an isothermal compression:

S − S0 = kbN lnVV0

, (3.26)

where N is the total number of particles. It is thus quite tempting to identify the two expressions andto say that the radiation is indeed composed of elementary particles, light quanta or photons, whosenumber at frequency ν is such that Nkb = Ekb/hν. The energy of each particle is thus E/N = hν.This sets a clear physical meaning to the Planck’s ansatz.

In the last paragraph of this paper, Einstein uses the light quantum concept to shed light onthe photoelectric effect, whose theoretical interpretation was also impossible in the frame of classicalelectromagnetism. This led him to the Nobel ceremony, in 1922. This historical paper marks the realbeginning of quantum theory. We will now use the modern formalism to quantify properly the field.

3.2 Field eigenmodes

As shown in the introductory section, the first step in field quantization is to identify properly a setof orthogonal modes on which any solution of the Maxwell equations can be expanded. We will dothat in this Section is a quite systematic way. This is a section on classical electromagnetism, but thework done here will make most of the quantization a trivial process. We first define the modes andthen turn to the field energy. The last paragraph will be devoted to the field momentum and angularmomentum.

3.2.1 Mode decomposition of the electromagnetic field

We set up some limiting conditions. This can be a fictitious box, as the one we used for the Planckcalculus, or a real cavity with mirrors surrounding an empty space. The electric field E(r, t) and themagnetic field B(r, t) obey Maxwell equations and the limiting conditions. We are interested in thefield only and there will thus be no sources in the box. We shall use the Coulomb gauge, and theelectric potential is thus identically zero. Of course, it is of interest to introduce the Fourier transformsof the fields.


Positive frequency field

We thus define the elecric field time Fourier transform by

E(r, t) =1√2π

∫ ∞−∞

E(r, ω)e−iωt dω . (3.27)

Since E is a real field,E∗(r, ω) = E(r,−ω) . (3.28)

We can thus split the Fourier transform in two parts and define the ‘positive frequency field’, integralover only positive frequencies:

E+(r, t) =1√2π

∫ ∞0

E(r, ω)e−iωt dω . (3.29)

and the ‘negative frequency field’:

E−(r, t) =1√2π

∫ 0

−∞E(r, ω)e−iωt dω =

(E+(r, t)

)∗. (3.30)

Note that the positive frequency field is not real in the general case. The real field is the sum of itsparts

E(r, t) = E+(r, t) + E−(r, t) . (3.31)

This splitting into positive and negative frequency components can of course be extended to all relevantfields and potentials.

Eingenmode basis

We thus set a ‘box’ of limiting conditions with a total volume V. These conditions can be thecancellation of field components (reflecting boundary conditions) or, as above, periodic boundaryconditions (more appropriate for practical use). Inside the box, we look for a solution of the Maxwellequation. All fields are divergence-free (no sources).

Mathematicians show that the ensemble of solutions for the electric field with limiting conditionsform a Hilbert space, on which we can always define an orthogonal basis. The basis vectors areharmonic and evolve with a positive frequency1. They can be written as:

f`(r)e−iω`t , (3.32)

where the dimensionless amplitude f` is divergence-free and obeys the Helmholtz equation:

∆f` +ω2`

c2f` = 0 . (3.33)

The index ` enumerates all these solutions that we will call ‘modes’ from now on. Of course, ` maybe a set of indices and not a single integer (see the Planck calculation above). Note also that thefrequencies ω` are generally quantized due to the limiting conditions imposed onto the f`s.

This mode basis can be chosen to be orthogonal:∫Vd3r f∗` (r) · f`′(r) = δ`,`′V , (3.34)

the normalization condition being: ∫Vd3r |f`(r)|2 = V . (3.35)

1With the simple relation between the positive and negative frequency parts of the field, we need only to expand iton a basis of positive frequency solutions.


We have used here a standard scalar product definition for complex vector functions. We operate ina finite volume. All integrals are thus convergent if the functions are not singular. Note that othernormalization choices would be possible. For instance, when studying cavity quantum electrodynamicsin Chapter 4, we will normalize the modes with another condition, better adapted to a problem withreal limiting conditions. This normalization choice only entails numerical factors here and there anddo not, of course, alter the final physical results.

The positive frequency field can be expanded onto this basis according to:

E+(r, t) =∑`

E`(t)f`(r) , (3.36)

where the amplitude E` (which has the dimension of an electric field) can be written, using the modebasis orthogonality as:

E`(t) =1

V

∫E+(r, t) · f∗` (r) d3r . (3.37)

Of course the time evolution of the amplitude E` is harmonic:

E`(t) = E`(0)e−iω`t . (3.38)

This results directly from the definition of the basis. It can also be recovered by plugging the expansionof E+ into the d’Alembert equation for the field and observing that f` obeys the Helmholtz equation.We leave this as an exercise to the reader. Finally,

E+(r, t) =∑`

E`(0)e−iω`tf`(r) . (3.39)

Plane wave basis

Let us now give a practical example of mode decomposition. We consider, as in the introductorySection, a rectangular box with dimensions Lx, Ly and Lz and chose periodic boundary conditions.A set of solutions of the Maxwell equations is made up of plane waves with a wavevector indexed bya set of three integers n = (nx, ny, nz) kn = (kx, ky, kz) = (nx2π/Lx, ny2π/Ly, nz2π/Lz).

For each set of indices n we must chose two orthogonal polarizations ε1 and ε2, perpendicular to k.In most cases, we shall take linear polarizations. The first polarization ε1 can be constructed by takingthe vector product of k with a fixed vector (not collinear with any k) and by imposing ε1 × ε2 = uk,where uk is the unit vector in the direction of k.

The index ` has thus four components: ` = (nx, ny, nz, ε) where ε = 1, 2. With these notations,the basis vectors are:

f`(r) = ε`eik`·r , (3.40)

which have modulus one, ensuring that the normalization condition∫|fell|2 = 1 is automatically

fulfilled. The positive frequency field is written as:

E+(r, t) =∑`

E`(0)ε`e−iω`teik`·r , (3.41)

a sum of plane waves.In a few instances (discussion of the field angular momentum) it is fruitful to use a basis of circularly

polarized waves. The corresponding unit vectors can be defined from the linear basis by;

ε± =ε1 ± iε2√

2, (3.42)

with the obvious relation ε+ = ε∗−. We introduce one useful relations which can be obtained byexpanding the circular polarizations on the linear ones in the vector products and using the linearpolarizations geometry:

ε+ × ε− = −iuk . (3.43)


Mode basis change

The mode basis is not determined in a unique way. For instance, the basis for the polarizations canbe changed, we can turn a plan wave basis into a standing wave basis, or we can use Gaussian lasermodes instead of plane waves. We will consider in this paragraph such basis changes. We have twosets of mode functions, f` and gp. The ‘old’ basis f` can be expanded on the other one according to

f` =∑p

U`pgp . (3.44)

We should not forget at this point that the modes have well defined frequency. We shall thus restrictourselves to a situation in which the modes coupled by the basis change have the same frequencies(if the sum runs on all modes, all elements U`p connecting modes with different frequencies should bezero).

The base orthonormality allows us to write:

U`p =1

V

∫f` · g∗p d3r . (3.45)

The basis change matrix must be a unitary: U †U = 11. We can check that by an explicit calculationas an exercise. Let us write the scalar product of two f modes:

δ`,`′ =1

V

∫f∗` · f`′ d3r =

∑p,p′

U∗`pU`′p′1

V

∫g∗p · gp′ d3r . (3.46)

Using the orthonormality of the g basis, we get

δ`,`′ =∑p

U∗`pU`′p =∑p

U`′pU†p` , (3.47)

and hence 11 = UU † as expected.

3.2.2 Normal variables

We need now to chose a simple set of dynamical variables to describe the evolution of all fields and interms of which we can write the system’s Hamiltonian. We will use for that purpose the coefficientsof the mode expansion of the potential vector.

Potential vector

The potential vector A is divergence free and, in the Coulomb gauge, E = −∂A/∂t. It is thus, forharmonic modes, proportional to the electric field and can be used as a convenient dynamical variable.We also expand A in a positive and a negative frequency part and expand the positive part on thesame mode basis as E (they obviously verify the same limiting conditions and the same Helmholtzequation):

A+(r, t) =∑`

A`(t)f`(r) . (3.48)

Of course, for the free field we are considering here, the A` function evolve at the mode frequency ω`.The set of the scalar A(t) functions will be called the ‘normal variables’. We show now that all fieldsand relevant quantities can be expressed as a functions of them.

For energy calculations, it is preferable to use real fields. We will thus define the real and imaginaryparts of A` by:

A`(t) = A`(0)e−iωt = x`(t) + ip`(t) . (3.49)


All fields

From E+ = −∂A+/∂t (it is easy to show that the relation holds separately for the positive andnegative frequencies), we get:

E`(t) = −dA`dt

= iωÀ` , (3.50)

and henceE+(r, t) =

∑`

iωÀ`(t)f`(r) . (3.51)

The magnetic field is the rotational of A. Hence

B+(r, t) =∑`

A`(t)h`(r) , (3.52)

whereh`(r) = ∇× f`(r) . (3.53)

We note immediately that the mode basis for the magnetic field is not the same as for the electricfield. Obviously, both fields share the same propagation equation and, when harmonic, obey the sameHelmholtz equation. However, the limiting conditions are generally quite different (for instance onconductors, where the tangential component of the electric field vanishes while that of the magneticfield is maximum). The passage from f to h is not a mode basis change! The electric potentialvanishes. We have now expressed all fields in terms ot the normal variables and we can deduce theexpression of the field energy, which will be the field Hamiltonian in the quantum realm.

3.2.3 Field energy: canonical variables

The total field energy H is simply given by:

H =ε02

∫E2 +

1

2µ0

∫B2 . (3.54)

Let us write it in terms of the normal variables.The energy must be written in terms the real fields, for instance:

E = 2Re E+ = 2Re∑`

iωÀ`f` , (3.55)

and a similar expression for B. When inserting these mode sums in the energy and expanding thesquares, we will get two types of terms. A first class involves scalar products of different field modefunctions. They identically cancel since we use and orthonormal mode basis. Only the square termsremain, which imply one and the same mode. It means that the total field energy is the direct sum ofthe energies of each mode considered separately, a considerable simplification:

H =∑`

H` . (3.56)

We are left with the much simpler problem to compute the total energy stored in a single modewith index `. In order to make the notations more compact, we will for a time omit the index ` in thecalculations, since it is shared by all quantities.

Let us start with the electric energy He. The real field is

E = iω [Af −A∗f∗] . (3.57)

Separating the real and imaginary parts of A and introducing those of f defined by:

f = f ′ + if ′′ , (3.58)


we get, after a little bit of algebra:E = −2ω

[xf ′′ + pf ′

]. (3.59)

Inserting that in the expression of He and noting that the x and p variables are independent of r, weget finally

He = 2ω2ε0

[x2∫

(f ′′)2 + p2∫

(f ′)2 + 2xp

∫f ′ · f ′′

]. (3.60)

Let us now proceed with the magnetic energy Hb. In the same vein, we write

B = Ah +A∗h∗ = 2xh′ − 2ph′′ , (3.61)

where we have also separated h in its real and imaginary parts. Plugging in the energy, we get:

Hb =2

µ0

[x2∫

(h′)2 + p2∫

(h′′)2 − 2xp

∫h′ · h′′

]. (3.62)

The expressions of the electric and magnetic energies are very similar and it is quite tempting togroup them. In addition we are in classical electromagnetism and we know that, in free space, theelectric and magnetic average energies are the same. There must be thus simple relations between theintegrals of f and those of h. We need a bit of vector calculus to check that.

Let us start with the integral of (h′)2, with h = ∇× f . Reminding ourselves that:

∇ · (a× b) = b · (∇× a)− a · (∇× b) , (3.63)

we can write∇ · [f ′ × (∇× f ′)] = (∇× f ′)2 − f ′ · (∇×∇× f ′) . (3.64)

Since f ′ is, as f , divergence-free, the double rotational in the r.h.s. reduces to the opposite of theLaplacian of f ′, and, thanks to the Helmholtz equation verified by f ′ (both the real and imaginaryparts of f verify this equation), we get

∇ · [f ′ × (∇× f ′)] = (h′)2 − ω2

c2(f ′)2 . (3.65)

Integrating over the whole space (beyond the box used to defined the modes), we cancel the divergenceterm since the mode functions cancel identically outside the box, and we get finally:∫

(h′)2 =ω2

c2

∫(f ′)2 (3.66)

a remarkably simple result. The same derivation applies without modification to the imaginary partof the f and h functions: ∫

(h′′)2 =ω2

c2

∫(f ′′)2 (3.67)

The last term we must examine is∫

h′ · h′′. We start in a similar way from the vector identity:

∇ · [f ′ × (∇× f ′′)] = (∇× f) · (∇× f ′′)− f ′ · (∇×∇× f ′′) . (3.68)

The double rotational transforms as above and the integral over the whole space of the divergencecancels as well. After a little work, we find:∫

h′ · h′′ = ω2

c2

∫f ′ · f ′′ . (3.69)

We can thus rewrite the magnetic energy as:

Hb = 2ω2ε0

[x2∫

(f ′)2 + p2∫

(f ′′)2 − 2xp

∫f ′ · f ′′

]. (3.70)


When summing with the electric energy, we observe that the xp terms cancel. Using the normalization,∫(f ′)2 +

∫(f ′′)2 = V , (3.71)

we get finally:

H = 2ω2ε0V[x2 + p2

]. (3.72)

This is obviously the energy of a one-dimensional harmonic oscillator, where the real and imaginaryparts of the normal variable A play the role of the position and momentum respectively. Quantizingthe field is as simple as quantizing an harmonic oscillator!

Of course, when returning to the full multi-mode problem, we get a collection of independentoscillators and a total energy

H =∑`

H` =∑`

2ω2` ε0V

[x2` + p2

`

]. (3.73)

We must perform a last step to put the problem in terms that can be directly quantized. We needx and p variables (we drop again the mode index), which are canonically conjugate quantities (theirPoisson bracket is one and their commutator in the quantum realm will be ih).

For this, we need to perform a little bit of normalisation (we could have made the right choicesbefore, but they had looked very artificial at the time). The proper canonical (in the meaning ofanalytical mechanics) variables xc and pc should verify the Hamilton equations of motion:

dxcdt

=∂H

∂pcand

dpcdt

= −∂H∂xc

. (3.74)

The variables x and p are not canonical, since

dx

dt= ωp 6= ∂H

∂p= 4ω2ε0Vp . (3.75)

We thus define a canonical amplitude for each mode

α(t) = 2√ε0ωVA(t) (3.76)

and the canonical position and momentum as its real and imaginary parts:

α(t) = xc + ipc , (3.77)

i.e.

xc = 2√ε0ωVx and pc = 2

√ε0ωVp . (3.78)

In these terms, the mode energy is

H =ω

2

[x2c + p2

c

]. (3.79)

Note that the xc and pc coordinates are not dimensionless. They are homogeneous to the squareroot of an action, i.e. to the square root of the Planck’s constant. They are thus not in directcorrespondence with the position and momentum of a mechanical harmonic oscillator. This is only amatter of normalization choices and of numerical factors.

It is easy now to check that the canonical position and momentum indeed verify the Hamiltonequations of motion (3.74). They are thus the right variables to perform the canonical quantizationof the field. Before performing this quantization, we will remain classical for a moment and examinethe field momentum and angular momentum.


3.2.4 Field momentum and angular momentum

Total momentum

That an electromagnetic field carries a momentum is evident, particularly in the relativistic frame asexpressed by the Maxwell stress tensor. The density of momentum is found to be proportional to thePoynting vector:

g =Π

c2with Π =

E×B

µ0. (3.80)

Note that this proportionality is evident in terms of photons. Each photon having an energy hν musthave a momentum hν/c since it propagates at light velocity. Hence, the density of momentum isdirectly proportional to the flux of energy.

In order to make the calculations, we shall use the plane wave mode basis. This is the most adaptedto describe the momentum, since all photons in a plane wave have the same direction and hence thesame momentum. We therefore write:

E+(r, t) =∑`

E+` =

∑`

iω`A`(t)ε`eik`·r , (3.81)

andB+(r, t) =

∑`

B+` =

∑`

A`(t)(ik` × ε`)eik`·r , (3.82)

since the rotational of f` is trivial in the plane wave case.We inject these expressions in the Poynting vector and sum over all space to find the total momen-

tum P of the field2. There are in the expansion square terms involving a single mode and cross termsbetween two modes. When the two modes have different k wavevectors, it is clear that the integralover space cancels.

Let us examine in more details the cross terms between the two polarization modes associated tothe same wavevector. Assuming linear polarizations for the sake of simplicity, the first mode has anelectric field along ε1 and thus a magnetic field along ε2. This is reversed for the other mode. Theelectric field of one mode is always aligned with the magnetic field of the other, canceling the crossterms in the Poynting vector.

The total momentum is thus the sum of the momenta stored in all modes: P =∑` P` with:

P` = ε0

∫(E+

` + E−` )× (B+` + B−` ) . (3.83)

We then plug the expression of the fields and expand all the terms. Some contain phase factors ofthe form exp(2ik` · r). They integrate to zero. Only the terms independent of r can survive in theintegration. This computation is painful and we only give the final result:

P` = 2ε0Vω`|A`|2ε` × (k` × ε`) . (3.84)

The double vector product is simply equal to k` since the polarizations are transverse. Finally,introducing the canonical amplitude α` (whose modulus square has the dimension of h) we can write

P =1

2

∑`

|α`|2k` . (3.85)

The interpretation of this expression is simple. The momentum of a plane wave mode is in the directionof its wavevector and its is proportional to the square modulus of the canonical amplitude. Note thatthis square modulus is a constant of motion. The momentum is thus a constant of motion, a logicalresult for an isolated system.

2We use the same notation, P, for the total field momentum and the density of polarization in matter. There is littlerisk of confusion.


Angular momentum

We now examine the field’s total angular momentum. The task is more demanding and we will notcomplete it fully. The density of angular momentum with respect to the origin is simply r×g and wecan write the total angular momentum J as:

J = ε0

∫r× (E×B) d3r . (3.86)

Simplifying this integral is quite difficult and we will only give the final result in the Coulombgauge:

J = L + S , (3.87)

where

S = ε0

∫E×A d3r (3.88)

is called the field’s ‘intrinsic angular momentum’ and

L = ε0

∫d3r

∑j

Ej(r · ∇)Aj , j = (x, y, z) , (3.89)

is the field’s ‘orbital angular momentum’. The reference to the electron angular momentum in thesedenominations is obvious, but we should not forget we are here treating a classical field.

Let us discuss a bit the intrinsic angular momentum S, since its expression is relatively simple.We perform a plane wave mode expansion, but choose the circular polarization basis (for reasons thatwill be evident soon). When expanding the modes sum, the cross terms with different wavevectorscancel. However, it is necessary to keep all terms with the same k and the two circular polarizations.We thus distinguish the index n enumerating the wavevector and the ± index of the polarizations.One notes that the cross terms between the two circular polarizations also cancel, due to the relationε+ × ε∗− = 0 demonstrated above. After some work, we arrive at

S = iε0V∑n

ωn[An+A∗n+(ε+ × ε∗+) +An−A∗n−(ε− × ε∗−)− c.c.

], (3.90)

where we have used the fact that the frequency of the mode is independent of the polarization.Using now ε+ × ε∗+ = ε+ × ε− = −iuk and its complex conjugate ε− × ε∗− = iuk, and introducing

again the canonical amplitude, we arrive at a simple form:

S =1

2

∑n

[|αn+|2 − |αn−|2

]uk . (3.91)

The interpretation of this form is far simpler than the calculation leading to it. Each circularly po-larized mode contribute to the intrinsic angular momentum, by a quantity aligned with the wavevectorand proportional to the mode intensity. The right-circular polarized modes have a positive contribu-tion, the left-handed ones a negative contribution. We guess that the interpretation of this result interms of the intrinsic angular momentum of the light quanta will be straightforward.

Of course, the modes intensities are constant and the intrinsic angular momentum is also a constantof motion. Since J should be constant as for any isolated system, L is bound to be constant too.

This is almost the only property of L that we can deduce with simple calculations. Expandingit over the mode basis is extremely cumbersome. The orbital momentum depends upon the precisespatial configuration of the field. It is in particular non-zero for laser beams tailored to have anhelical phase, with a potential vector of the form U(r, z) exp(imφ) exp(i(kz − ωt)) in the cylindricalcoordinates r, φ, z (with of course U(r = 0) = 0). The phase rotation around the axis confers angularmomentum to these modes, and L = (H/ω)muz. Here also the interpretation in terms of photons willbe simple, each photon in this beam having an orbital angular momentum mh.


3.3 Field quantization

We can now take the final step and quantize the field. Each mode being totally independent from theothers, we can treat their quantization separately. We will thus drop the mode index ` when thereis no possible confusion. We assume of course that the quantization of a one-dimensional harmonicoscillator is well known.

3.3.1 Creation and annihilation operators, Hamiltonian

The classical field energy of one mode is, within a factor ω/2, equal to x2c + p2

c , where xc and pcare conjugate canonical variables. The quantization of such a ‘mechanical’ problem dates from thevery early days of quantum mechanics. The conjugate classical variables should be replaced by twooperators X and P (position and momentum operators) acting in an infinite dimension Hilbert space,with the commutation rule:

[X,P ] = ih . (3.92)

From X and P , we can build quantum analogues of the complex classical oscillation amplitude,proportional to xc + ipc. Let us thus define the non-hermitian operator a by:

a =1√2h

(X + iP ) , (3.93)

whose ajoint is

a† =1√2h

(X − iP ) , (3.94)

It is easy to compute the commutator of a with a† from the commutation relation of X and P :[a, a†

]= 11. (3.95)

We can invert the definition of a and a† and get

X =

√h

2

(a+ a†

), (3.96)

and

P = i

√h

2

(a† − a

). (3.97)

It is often useful to introduce reduced units for the position and momentum operators. We thusdefine here the field quadratures X0 and P0 by:

X0 =X√2h

and P0 =P√2h

. (3.98)

With these definitions:

[X0, P0] =i

2(3.99)

a = X0 + iP0 , a† = X0 − iP0 , X0 =a+ a†

2, P0 = i

a† − a2

. (3.100)

The quantum mode Hamiltonian is in direct correspondence with the classical one, by replacingthe canonical variables by the position and momentum operators (no commutation problem occurs inthis replacement. Hence

H =ω

2(X2 + P 2) = hω(X2

0 + P 20 ) . (3.101)

3.3. FIELD QUANTIZATION 79

Let us express it in terms of the a and a† operators:

H =hω

4

[(a+ a†)2 − (a† − a)2

]. (3.102)

The square terms cancel. Using the commutation relation of a and a† to put the double products inthe so-called ‘normal order’ (all a†s on the left of the as, we finally arrive at

H = hω

(a†a+

1

2

). (3.103)

Diagonalizing this Hamiltonian is a simple procedure, described in all textbooks. It amounts dodiagonalizing the ‘number operator’

N = a†a , (3.104)

which has the following commutation relations:

[N, a] = −a and[N, a†

]= a† . (3.105)

It is easy to show that the spectrum of N is the set of positive or zero integers. The eigenstates of Ndefined by

N |n〉 = n |n〉 , (3.106)

are non-degenerate and called the Fock states. They are also the eigenstates of the Hamiltonian Hwith an eigenvalue

En =

(n+

1

2

)hω . (3.107)

The ground state is the ‘vacuum state’, |0〉, with a non-zero hω/2 energy. This ‘vacuum energy’ isof course a totally non-classical effect linked to the Heisenberg uncertainty relations.

The Fock states form an equidistant ladder of levels, separated by the energy of a photon hω. Wecan thus interpret the Fock states as states with a well-defined number of photons. They are thusalso called ‘photon-number states’. Of course, the Fock states are an orthonormal basis in the Hilbertspace with:

〈n |p〉 = δn,p . (3.108)

The a operator transforms |n〉 into |n− 1〉 according to:

a |n〉 =√n |n− 1〉 , (3.109)

and, similarly

a† |n〉 =√n+ 1 |n+ 1〉 , (3.110)

The operator a is thus called the ‘annihilation’ operator since it removes one photon, and a† is calledthe ‘creation’ operator. All Fock states can thus be generated from the vacuum by:

|n〉 =(a†)n√n!|0〉 . (3.111)

Note also that

a |0〉 = 0 , (3.112)

an essential property of the vacuum state.Let us now return to the full multi-mode case. The total Hilbert space is the tensor product of

all spaces associated to the separate modes. Its dimension is thus quite infinite. Since the modesare dynamically uncoupled, the operators of one mode commute with all the other modes operators.


The vacuum |0〉 is the tensor product of the vacuum states of all modes. The eigenstates of the totalHamiltonian H are the tensor products of Fock states of each mode:

H |n1, . . . , n` . . .〉 = En |n1, . . . , n` . . .〉 , (3.113)

with

En =∑`

(n`hω` +

hω`2

). (3.114)

and

|n1, . . . , n` . . .〉 =∏`

(a†`)n`

√n`!|0〉 . (3.115)

We note that the vacuum has an infinite energy. This is the harbinger of a considerable difficulty inthis simple approach to field quantization. The zero-point energy is quite harmful for some problems,leading to infinities in many calculations, such as the atomic energy level shifts due to the continuumof empty field modes (Lamb shift effect). We will discuss this matter a bit further in the next Chapterand see how we can in most cases circumvent the difficulty without getting in the technically complexrenormalization procedures of QED.

3.3.2 Field operators

We now proceed to express all the relevant field operators in terms of the annihilation and creationoperators, in correspondence with the classical fields. The normal variables for the classical fields arethe vector potential amplitudes A, which are expressed in terms of the canonical variables by:

A =1

2√ε0ωV

(xc + ipc) . (3.116)

The corresponding quantum mechanical operator is thus, reintroducing the modes indices:

A` =1

2√ε0ω`V

(X` + iP`) =

√h

2ε0ω`Va` , (3.117)

and is simply proportional to the annihilation operator. We just need to replace, within this factor,in all the classical fields A` by a` and A∗` by a†`.

Fields

The positive-frequency vector potential is now a non-hermitian operator in the field Hilbert space anddepends upon the classical position r:

A+(r) =∑`

√h

2ε0ω`Va`f`(r) . (3.118)

The hermitian full vector potential is the sum of A+ and of its complex conjugate:

A(r) =∑`

√h

2ε0ω`V

(a`f`(r) + a†`f

∗` (r)

). (3.119)

The hermitian electric field is similarly:

E(r) = i∑`

E`(a`f`(r)− a†`f

∗` (r)

), (3.120)


where we define the ‘field per photon in mode `’ by

E` =

√hω`2ε0V

. (3.121)

The magnetic field is:

B(r) =∑`

√h

2ε0ω`V

(a`h`(r) + a†`h

∗` (r)

), (3.122)

with h` = ∇× f`.In all these expressions, all the details of the field geometry (mode structure, polarization...) are

encapsulated in the classical mode functions. The only quantum parts are the creation and annihilationoperators. A photon can thus be defined as an elementary excitation of a classical field mode, solutionof Maxwell equations with limiting conditions. As such, a photon has a polarization, a mode structure.It can inhabit a plane wave, a standing wave pattern or a propagating Gaussian beam. In this pointof view, of course, the photon cannot be pictured as a localized particle. It is everywhere in the modeat the same time (the relative mode amplitude modulus square giving the probability to detect thephoton at a point – see the paragraph on photodetection in Chapter 4).

If we use the plane-wave basis, we can write (we give only the positive frequency parts of the fieldoperators to save space):

A+(r) =∑`

√h

2ε0ω`Va`εè

ik`·r , (3.123)

E+(r) = i∑`

Eà`εèik`·r , (3.124)

B+(r) =∑`

√h

2ε0ω`Va`(ik` × ε`)eik`·r . (3.125)

Note that there is no explicit time dependence, since we have replaced the time-dependent ampli-tudes by operators. We are in the Schrodinger picture and all time dependence is encapsulated in thestate. We can recover time-varying fields, close to the classical picture, by switching to the Heisenbergrepresentation. The evolution equation for aH , the annihilation operator in the Heisenberg picture is

ihdaHdt

= [aH , H] i.e.daHdt

= −iωaH , (3.126)

whose immediate solution isaH(t) = aH(0)e−iωt = ae−iωt . (3.127)

Substituting this in the field operators, we get the fields in the Heisenberg picture, which have thesame time dependence as their classical counterparts.

Momentum, angular momentum

The momentum3 has been expressed in terms of the canonical classical amplitude αl, which is pro-portional to A`. Its quantum counterpart is just a`

√2h. Substituting |α`|2 by α∗`α` in the expression

of the momentum, we getP =

∑`

hkl a†à` . (3.128)

3The momentum of the field and the momentum associated to the fictitious unidimensional harmonic oscillatorequivalent to a field mode are not related at all!


Note that we have an ordering problem for a and a† in the transformation of the quadratic classicalquantity into a quantum operator. We have here made implicitly the only reasonable choice, that ofnormal ordering, since the momentum cancels in the vacuum state for this form only. Changing theordering would add an infinite constant terms (

∑` hk`) to the momentum.

The interpretation of this expression of the momentum is evident. Each photon in the plane-wavemode ` has a momentum hk`, the modulus of which is hν/c, and the total momentum is the sum ofall the photons momenta.

The same simple reasoning applies to the intrinsic angular momentum of the field. We get:

S =∑n

hukn [Nn+ −Nn−] . (3.129)

each right (left) circularly polarized photon has an intrinsic angular momentum h (−h) along its direc-tion of propagation. The photon thus appears as a spin-1 particle. However, the photons never havea zero intrinsic angular momentum along their direction of propagation, an immediate consequence ofthe transverse nature of the electromagnetic waves. A linearly polarized wave corresponds to photonsin a quantum superpositions of having a ±h angular momentum.

Field quadratures

Let us return to the two field quadratures X0 and P0. We can define their eigenstates which correspondto the position or momentum eigenstates of the harmonic oscillator:

X0 |x〉 = x |x〉 and P0 |p〉 = p |p〉 . (3.130)

Note that these states, which are a basis of the Hilbert space, cannot be properly normalized, astandard pathology in infinite dimensional Hilbert spaces. The position x and momentum p define aclassical phase space for the mode’s harmonic oscillator, equivalent to the Fresnel plane for the electricfield.

With these states, we can define wavefunctions for the field either in the X0 or P0 representation.For instance, let:

Ψ(x) = 〈x |Ψ〉 . (3.131)

The wavefunction of the vacuum state |0〉 is found to be

Ψ0(x) =

(2

π

)1/4

e−x2

(3.132)

a simple Gaussian with a width 1. The wavefunction of the same state in the P0 representation isthus the same Gaussian, since both wavefunctions are linked by a Fourier transform relation. We canthus view the ground state in the phase space as a unit radius blob centred at the origin. We will givelater a much more precise representation of this fact.

The Fock state wavefunction is then:

Ψn(x) =

(2

π

)1/4 1√2nn!

e−x2Hn(x

√2) , (3.133)

where Hn is the nth Hermite polynomial defined by

Hn(u) = (−1)neu2 dn

dune−u

2. (3.134)

These wavefunctions have n− 1 nodes and a a parity (−1)n. They are plotted in all textbooks.We define now a more general field quadrature by:

Xφ =ae−iφ + a†eiφ

2, (3.135)


with the evident identification of X0 with the φ = 0 quadrature. We also have Xπ/2 = P0. It can beeasily shown that: [

Xφ, Xφ+π/2

]=i

2, (3.136)

resulting in the Heisenberg uncertainty relations

∆Xφ∆Xφ+π/2 ≥1

4. (3.137)

The eigenstates of the quadratures are defined by Xφ |xφ〉 = xφ |xφ〉 and are linked by Fourier trans-formations; ∣∣∣xφ+π/2

⟩=

1√π

∫dyφe

2ixφ+π/2yφ |yφ〉 (3.138)

Mode basis change

We now examine the quantum version of the mode basis change described in the last Section. Wechange from the mode functions f` to the mode functions gp, with

f` =∑p

U`pgp , (3.139)

where U`p is a unitary matrix that connects only the modes with identical frequencies.The positive frequency part of the electric field can be written as:

E+ = i∑`

E`f`(r)a`

= i∑`,p

EÙ`pa`gp(r)

=∑p

Epgp(r)bp , (3.140)

withbp =

∑`

U`pa` , (3.141)

The last relation defines the annihilation operator bp of the ‘new’ modes as a linear combinationof the ‘old’ modes annihilation operators. A mode basis change thus reflects on the field operators.Note that the fact that U`p connects only modes with the same frequency plays an essential role inthis computation, when it comes to identifying the fields per photon E` and Ep. The conjugate of the

previous equation leads, since U∗`p = U †p`, to

b†p =∑`

U †pà†` , (3.142)

where we make use of U∗`p = U †p`. As an exercise, we can check that the bosonic commutation relationsof the annihilation and creation operators is preserved in this transformation:[

bp, b†q

]=

∑`,m

U`paÙ†qma

†m − U †qma†mU`pa`

=∑`,m

U`pU†qm

[a`, a

†m

]=

∑`

U †qÙ`p

= δp,q , (3.143)

where we take care that, for the field modes, the matrix U is made up of numbers, not of operators

and where we use[a`, a

†m

]= δ`,m.


3.4 Field quantum states

We now examine in more details the field quantum states. We have already introduced the Fock states,eigenstates of the field Hamiltonian. We will show that they are utterly non-classical in nature. Wewill thus introduce also the coherent states,

3.4.1 Fock states

We have already given most of the properties of these eigenstates of the field Hamiltonian. They area convenient basis of the Hilbert space and any field state can be written as:

|Ψ〉 =∑n

cn |n〉 . (3.144)

The ‘photon number distribution’ for state |Ψ〉, pn, i.e. the probability for finding n photons in anideal photon counting experiment is:

pn = |cn|2 . (3.145)

The mean number of photons in state |Ψ〉 is thus:

n =∑n

npn , (3.146)

and the photon number variance is:

∆N2 =⟨N2⟩− 〈N〉2

=∑n

(n− n)2 pn . (3.147)

The average values of a and a† in a Fock state vanish, since a |n〉 and a† |n〉 are both orthogonal to|n〉. It results that the average value of all fields cancel in a Fock state. This reveals the non-classicalnature of these states. They have a finite, possibly large energy, but they correspond to zero electricand magnetic fields! This is due to the fact that the Fock states bear no phase information. In aclassical picture, they represent a field with a totally random phase.

In most cases, the field states are not pure states, but statistical mixtures. This is particularly thecase when field relaxation is involved, as described in the end of this Chapter. In this situation, thefield state must be described by a density operator:

ρ =∑n,p

ρnp |n〉〈p| . (3.148)

The diagonal elements of the density matrix correspond to the photon number distribution:

ρnn = pn . (3.149)

The non-diagonal elements describe the phase information contents of the state. We will examine theimportance of these coefficients in the next paragraph.

Let us mention finally that the Fock states with at least one photon are of course not invariantunder a change of the mode basis. The state with n photons in the mode bp can be expressed as:

|np〉 =(b†p)

np

√n!|0〉 =

(∑` U†p`a†`

)np√n!

|0〉 (3.150)

The total number of photons is the same in both cases, but the np photons stored in mode bp areshared by all the coupled a` modes. We will see a more practical example of this situation whentreating the beam-splitter later in this Chapter.

3.4. FIELD QUANTUM STATES 85

3.4.2 Coherent states

We introduce in this paragraph the coherent states of the field, which are much closer to the classicalpicture of non-zero oscillating electric and magnetic fields than the non-classical Fock states. Thesestates are quite important, since they are produced by all classical field sources (electronic oscillatorsor lasers) involving a large, coherent emitting system (either a classical oscillating current or a largecollection of atoms).

Definition

Let us first define the unitary ‘displacement’ operator:

D(α) = eαa†−α∗a , (3.151)

where α is an arbitrary complex amplitude, which can be split in its real and imaginary parts accordingto:

α = α′ + iα′′ . (3.152)

The displacement operator is obviously unitary :

D(α)†D(α) = 11 . (3.153)

We also note that

D(α)† = D(−α) . (3.154)

The coherent state |α〉 is defined as the action of the displacement operator on the vacuum:

|α〉 = D(α) |0〉 . (3.155)

Since D(0) = 11, we observe that the vacuum state |0〉 is also the coherent state with zero amplitude.Let us examine in more details the action of the displacement operator. We separate the real andimaginary parts of α and write

D(α) = e2iα′′X0−2iα′P0 . (3.156)

We now make use of the Glauber relation:

eAeB = eA+Be[A,B]/2 , (3.157)

valid when both A and B commute with their commutator:

[A, [A,B]] = [B, [A,B]] = 0 . (3.158)

It thus applies above since [X0, P0] = i/2 is proportional to the identity. We thus get:

D(α) = e−iα′α′′e2iα′′X0e−2iα′P0 . (3.159)

The last operator is obviously a translation operator for the position of the harmonic oscillator, by anamount α′. The next operator is a translation in momentum by an amount α′′:

e−2iα′P0 |x〉 =∣∣x+ α′

⟩(3.160)

e2iα′′X0 |p〉 =∣∣p+ α′′

⟩. (3.161)

The coherent state appears thus, within a global phase factor, as the vacuum state translated byα′ in position and α′′ in momentum. This justifies fully the name ‘displacement operator’ coined for


D(α). This simple feature provides in a simple way the wavefunction of the coherent state in the X0

representation:Ψα(x) ∝ e−(x−α′)2

, (3.162)

and in the P0 representation:Ψα(x) ∝ e−(p−α′′)2

. (3.163)

Let us now examine the action of the displacement operator on the annihilation operator bycomputing D(α)aD(−α). Using the Baker-Hausdorff lemma:

eAae−A = a+ [A, a] +1

2![A, [A, a]] + . . . , (3.164)

for A = αa† − α∗a and noting that [A, a] = −α and, hence, that all higher order commutators cancel,we get immediately:

D(α)aD(−α) = a− α11 . (3.165)

The displacement operator thus performs a translation on the annihilation operator.We can also compute the combination of displacements. Using again the Glauber relations, it is

easy to show thatD(α)D(β) = e(αβ∗−α∗β)/2D(α+ β) . (3.166)

Within a phase factor, the displacement operators add as the classical fields. Let us interpret the phasefactor iΦ = (αβ∗ − α∗β)/2, where Φ is obviously a real phase. Separating the real and imaginaryparts of the displacements amplitudes, we get:

Φ =α′′β′ − α′β′′

2. (3.167)

This is precisely half the vector product of the two vectors α and β in the phase plane. The phase Φ isthus simply the surface of the triangle with sides α and β. This phase is of a geometrical, topologicalnature. This is a simple example of a Berry’s phase observed when a quantum state is displaced.

Properties of the coherent states

Let us examine now the main properties of the coherent states. First, they are right-eigenstates of theannihilation operator:

a |α〉 = aD(α) |0〉 = D(α)D(−α)aD(α) |0〉 = (a+ α11) |0〉 = α |α〉 , (3.168)

since a |0〉 = 0. The annihilation operator a being non-hermitian, there are no restriction on itsspectrum, which appears here as made of all complex numbers. Hence:

〈α| a† = α∗ 〈α| , (3.169)

and the coherent states are left-eigenstates of the creation operator. Note that there is no simpleexpression for a† |α〉.

This property entails that the annihilation and creation operators have an non-zero expectationvalue in the coherent states:

〈α| a |α〉 = α and 〈α| a† |α〉 = α∗ (3.170)

The average values of all fields are thus also non zero and, for instance:

〈E〉 = iE (f(r)α− f∗(r)α∗) (3.171)

〈A〉 =Eω

(f(r)α+ f∗(r)α∗) , (3.172)


a result very close to the expressions of the classical fields.

The average number of photons in the coherent state is simply:

n = 〈α| a†a |α〉 = |α|2 . (3.173)

We can also compute easily the variance of the photon number, using N2 = a†aa†a = (a†)2a2 + a†a.Hence: ⟨

N2⟩

= |α|4 + |α|2 , (3.174)

and

∆N2 = |α|2 = n . (3.175)

The variance of the photon number is equal to its average, a characteristic feature of the Poissondistribution. We observe that:

∆N

n=

1√n. (3.176)

The larger the energy of the coherent state, the less is the relative uncertainty on this energy and themore the coherent state looks like a classical field.

We can make this observation more precise by computing the expansion of the coherent states onthe Fock state basis. Using once again the Glauber relation to expand the displacement operator as:

D(α) = e−|α|2/2eαa

†e−α

∗a , (3.177)

and noting that e−α∗a |0〉 = |0〉 (in the power expansion of the exponential, all terms cancel but the

identity one), we get:

|α〉 = e−|α|2/2eαa

† |0〉 . (3.178)

We expand the exponential operator in power series and note that (a†)n |0〉 is proportional to |n〉. Wefinally get:

|α〉 =∑n

cn |n〉 , (3.179)

with

cn = e−|α|2/2 α

n

√n!

. (3.180)

This expression shows that |0〉 is indeed the only state which is, at the same time, a photon numberstate and a coherent state.

The photon number distribution of the coherent state |α〉 is thus given by:

pn = e−|α|2 |α|2n

n!= e−n

nn

n!, (3.181)

a Poisson distribution as expected. For very large average photon numbers, this distribution is nearlyGaussian, with

pn ∝ e−(n−n)2/n . (3.182)

The larger the photon number, the more peaked (in relative value) is this distribution. We recoverthe fact that a large coherent field has a well-defined energy, as a classical field.

We can now finally compute the scalar product of two coherent states:

〈α |β〉 = e−(|α|2+|β|2)/2∑n,p

(α∗)nβp√n!p!

〈n |p〉

= e−(|α|2+|β|2)/2eα∗β (3.183)


and its square modulus|〈α |β〉 |2 = e−|α−β|

2. (3.184)

Two coherent states are thus never orthogonal. Nevertheless, when the difference of their amplitudeshas a modulus much larger than one, they can be considered as nearly orthogonal for all practicalpurposes.

We have seen that the coherent states are displaced vacuum states. The vacuum can be visualizedas a radius unity blob around the origin of phase space. The coherent state |α〉 can thus be alsovisualized as a blob, centred on the complex amplitude α. The scalar product of |α〉 and |β〉 nearlycancels as soon as the two blobs no longer overlap.

These non-orthogonal states form an over-complete basis of the Hilbert space:

11 =1

π

∫d2α |α〉〈α| . (3.185)

In order to establish this important property, we expand in the integral the ket and the bra over theFock states basis: ∫

d2α |α〉〈α| =∑n,p

1√n!p!|n〉〈p|

∫d2α e−|α|

2αn(α∗)p . (3.186)

We compute the integral over the phase plane in polar coordinates, with α = ρ exp(iθ). The integralis then: ∫

ρdρdθ e−ρ2ρn+peiθ(n−p) , (3.187)

and obviously cancels when n 6= p. For n = p, the integral reduces, with u = ρ2, to:

In = π

∫duune−u . (3.188)

An integration per parts gives In = nIn−1. Since I0 = π, we get In = πn! and hence:∫d2α |α〉〈α| = π

∑n

|n〉〈n| , (3.189)

completing the demonstration.The coherent state basis being over-complete, the expansion of a quantum state onto it is not

unique. This is particularly the case for the vacuum. On the one hand, it is itself a coherent state.On the other hand, using the generalized closure relation above:

|0〉 =1

π

∫d2α e−|α|

2/2 |α〉 . (3.190)

Let us finally note that the Fock state can (non uniquely) be expanded on the coherent states as:

|n〉 =1

π√n!

∫d2α e−|α|

2/2(α∗)n |α〉 . (3.191)

Evolution

The coherent states are not eigenstates of the mode’s Hamiltonian. They thus evolve with time in anon-trivial way. Let

|Ψ(0)〉 = |α〉 = e−|α|2/2∑n

αn√n!|n〉 . (3.192)

Then

|Ψ(t)〉 = e−|α|2/2∑n

αn√n!e−inωte−iωt/2 |n〉

= e−iωt/2∣∣∣αe−iωt⟩ . (3.193)


Within a phase factor (that could be eliminated in a single mode problem by a convenient interactionrepresentation i.e. by choosing the vacuum energy as being zero), the coherent state remains coherentand the amplitude evolves as the classical complex field amplitude:

α(t) = α(0)e−iωt . (3.194)

The wavefunction of the coherent state is thus a Gaussian which oscillates around the originwithout deformation. Coherent states escape the dispersion of the wave-packet that affects free spacepropagation. Accordingly, all the average values of the field operators oscillate at the frequency ω astheir classical counterparts. The coherent state are thus clearly the ‘most classical’ states and retaintheir features throughout their evolution.

Let us conclude this paragraph by a note on the creation of the coherent states. We will treat itin details when we will have a proper interaction Hamiltonian for charges coupled to the field. Wewill show that a classical oscillating current, resonantly coupled to the mode, produces an interactionproportional to a + a†. It is clear that the evolution operator associated to this interaction is adisplacement. Thus, a classical oscillating current creates a coherent state. Not only are the coherentstates blessed with remarkable properties, but they are also the easiest to produce experimentally. Onthe contrary, the Fock states, excluding the vacuum of course, are rather difficult to obtain and haveonly been produced in relatively recent quantum optics experiments.

3.4.3 Phase space representations

Phase space distributions4 are fundamental tools in classical statistical physics. For a particle un-dergoing a one-dimensional motion, the phase space is a plane, with the position x and conjugatemomentum p as coordinates. A point in this plane defines the mechanical state of the particle. In thecase of a field mode, equivalent to a one-dimensional oscillator, the two orthogonal field quadraturesplay the role of position and momentum, and a point in the Fresnel plane defines a classical fieldstate. Statistical uncertainties or lack of knowledge about the system are accounted for by replacingthe Dirac distribution associated to this point by a positive and normalized density of probability,f(x, p), which takes non-vanishing values in a limited region of phase space. The statistical average oof any observable quantity, o(x, p), is given by:

o =

∫f(x, p)o(x, p) dxdp . (3.195)

The system’s evolution due to external forces and to the coupling with the environment is describedby a diffusion equation for f , relating its temporal and spatial derivatives.

The extension of this phase space representation to quantum states was discussed for the firsttime by Wigner in 1932. A difficulty has immediately arisen because Heisenberg uncertainty relationsforbid, even in the absence of any statistical indeterminacy, the precise and simultaneous determi-nation of conjugate variables. In other words, a fundamental quantum blurring adds to the classicaluncertainties. In spite of this problem, it remains possible to define a real phase space function for aquantum particle, which retains some of the essential characteristic features of the classical probabil-ity distribution. This quantum distribution is called the Wigner function W . Its definition and mainproperties are recalled in this Section.

We will see that W is naturally defined from its Fourier transform, the system’s symmetricalcharacteristic function Cs(λ). Introducing Cs will lead us to consider two simply related characteristicfunctions, Cn(λ) and Can(λ), and we will see that their Fourier transforms define two phase spacedistributions different from W , the Husimi-Q function and the Glauber–Sudarshan P distribution.The latter, which is highly singular, is of difficult mathematical use and we will not say much aboutit. Here, we will discuss the connexions between W and Q, compare them and explain why W is moreuseful than Q for the analysis of the quantum features of a system.

4This Section is reproduced from Haroche and Raimond, ‘Exploring the quantum’, OUP, 2006


Characteristic functions

The three simple characteristic functions depend upon the choice made for the ordering when expand-ing in power series a function of the annihilation and creation operators a and a†. In the ‘normal’order, all creation operators are placed to the left of the annihilation ones (the photon number op-erator N = a†a is in normal order). The ‘anti-normal’ order puts all the a’s on the left. Finally,the ‘symmetric’ order produces expressions in which all products of operators are symmetrized. Forinstance, the direct power series expansion of the displacement operator:

D(λ) = eλa†−λ∗a = 11 + λa† − λ∗a+ λ2a†2 − λλ∗(a†a+ aa†) + λ∗2a2 + . . . , (3.196)

is naturally in symmetric order.

We now introduce three characteristic functions of the state of the system corresponding to these

orders. The symmetric-order characteristic function, C[ρ]s (λ), is the average of D(λ) in the state

described by the density operator ρ:

C [ρ]s (λ) = 〈D(λ)〉 = Tr

[ρeλa

†−λ∗a]. (3.197)

The operator D(λ) being unitary, all its eigenvalues have a unit modulus. The modulus of its averageis thus always bounded by one:

|C [ρ]s (λ)| ≤ 1 . (3.198)

Note also the identity:

C [ρ]s (0) = Tr(ρ) = 1 . (3.199)

The upper bound of C[ρ]s (λ) is thus reached at the origin. In addition, C

[ρ]s (λ) obeys the conjugation

relation:

C [ρ]s (−λ) =

[C [ρ]s (λ)

]∗. (3.200)

Let us finally give the expression of the symmetric characteristic function for a pure state |Ψ〉:

C [|Ψ〉〈Ψ|]s = 〈Ψ|D(λ) |Ψ〉 . (3.201)

It is merely the overlap between |Ψ〉 and the state obtained by translating |Ψ〉 in phase space by D(λ).It is thus an autocorrelation function of the quantum state in phase space.

The normal- and anti-normal-order characteristic functions are likewise defined as:

C [ρ]n (λ) = Tr

[ρeλa

†e−λ

∗a], (3.202)

and:

C [ρ]an(λ) = Tr

[ρe−λ

∗aeλa†]

. (3.203)

These functions can be related to each other with the help of the Glauber identity. We obtainimmediately:

C [ρ]n (λ) = e|λ|

2/2C [ρ]s (λ) ; C [ρ]

an(λ) = e−|λ|2/2C [ρ]

s (λ) . (3.204)

Any couple of characteristic functions can thus be determined from the knowledge of the third one bya mere multiplication by e±|λ|

2/2 or e±|λ|2.

As a simple example, let us determine the characteristic functions for a coherent state |α〉. Wehave:

C [|α〉〈α|]s (λ) = 〈α|D(λ) |α〉 . (3.205)

Recalling that:

D(λ) |α〉 = exp[(λα∗ − λ∗α)/2] |α+ λ〉 , (3.206)


we get:

C [|α〉〈α|]s (λ) = e(λα∗−λ∗α)/2 〈α |α+ λ〉 = e−|λ|

2/2eα∗λ−λ∗α , (3.207)

a complex Gaussian function from which we derive:

C [|α〉〈α|]n = eα

∗λ−λ∗α , (3.208)

and:

C [|α〉〈α|]an = e−|λ|

2eα∗λ−λ∗α . (3.209)

The characteristic functions of a Fock state |n〉 are obtained in a similar way, by expanding the normal-ordered displacement operator in powers of a† and a, retaining terms with the same number of creationand annihilation operators which are the only ones to have non-vanishing expectation values in an |n〉state . The final result for Cs is:

C [|n〉〈|n〉|]s = e−|λ|

2/2Ln(|λ|2) , (3.210)

where Ln is the nth Laguerre polynomial:

Ln(x) =n∑p=0

(−1)pn!

p!2(n− p)!xp . (3.211)

Let us finally mention the symmetric-order characteristic function for a thermal field, obtained bysumming the Fock state result over a thermal distribution (more on the thermal field equilibrium inthe last Section of this Chapter):

ρth =∑n

nnth(nth + 1)n+1

|n〉〈n| , (3.212)

where nth is the average number of thermal photons. We get a remarkably simple result:

C [ρth]s (λ) = e−(nth+1/2)|λ|2 . (3.213)

The Husimi-Q distribution

The Q distribution is the Fourier transform of the anti-normal-order characteristic function:

Q[ρ](α) =1

π2

∫d2λ e(αλ∗−α∗λ)C [ρ]

an(λ) . (3.214)

This expression looks formidable, but in fact, Q[ρ](α) is simply related to the expectation value of thedensity operator ρ in state |α〉. To show this, let us write Q as:

Q[ρ](α) =1

π3Tr

[ρ

∫d2λd2β e(αλ∗−α∗λ)e−λ

∗a |β〉〈β| eλa†], (3.215)

where we have used the definition (3.203) of Can and the closure relation on coherent states:

1

π

∫d2β |β〉〈β| = 11 . (3.216)

The coherent state |β〉 being an eigenstate of exp(−λ∗a), Q reduces to:

Q[ρ](α) =1

π3Tr

[ρ

∫d2λd2β eλ

∗(α−β)−λ(α∗−β∗) |β〉〈β|], (3.217)


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02

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0

0 . 0 5

0 . 1

0 . 1 5

0

0 . 0 5

0 . 1

0 . 1 5

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02

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02

4

0

0 . 0 5

0 . 1

0 . 1 5

0

0 . 0 5

0 . 1

0 . 1 5

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02

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02

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0

0 . 1

0 . 2

0 . 3

0

0 . 1

0 . 2

0 . 3

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02

4

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02

4

0

0 . 0 2

0 . 0 4

0

0 . 0 2

0 . 0 4

( a ) ( b )

( d )( c )

Figure 3.1: Husimi-Q distributions. (a) Coherent state |β〉, with β =√

5. (b) Five-photon Fock state. (c) Schrodinger

cat state, superposition of two coherent fields |±β〉, with β =√

5. (d) Statistical mixture of the same coherent compo-

nents.

and the integration over λ leads to a Dirac distribution δ(α − β) according to the standard Fouriertransform5 relation:

1

π2

∫d2λ eλβ

∗−λ∗β = δ(β) . (3.218)

We thus finally get:

Q[ρ](α) =1

πTr [ρ |α〉〈α|] =

1

π〈α| ρ |α〉 , (3.219)

which can also be written as:

Q[ρ](α) =1

π〈0|D(−α)ρD(α) |0〉 =

1

πTr[|0〉〈0|D(−α)ρD(α)] . (3.220)

The Q function is thus the average of the projector onto the vacuum state, |0〉〈0|, in the field displacedin phase space by −α. Being the expectation value of an observable, it is thus a directly measurablequantity.

The Q distribution is positive, bounded by 1/π and normalized (∫d2αQ(α) = 1), this last property

resulting directly from eqn. (3.216). Using eqn. (3.219), it is easy to compute the Q functions of ourfavourite classical and non-classical states.

The Q distribution of a coherent state |β〉 is:

Q[|β〉〈β|](α) =1

π| 〈α |β〉 |2 =

1

πe−|α−β|

2. (3.221)

It is plotted in Fig. 3.1(a) for β =√

5. It is a Gaussian, with a width 1, reaching for α = β themaximum allowed value 1/π. Note that the pictorial representation of coherent states as an uncertaintydisk superposed on a classical amplitude corresponds to a cut at 1/eπ in the Q[|β〉〈β|](α) distribution.

Expanding |α〉 on the Fock state basis, it is easy to find the Q function for an n-photon Fock state:

Q[|n〉〈n|](α) =1

π

|α|2n

n!e−|α|

2. (3.222)

5That the integral written in this equation is a bidimensional Fourier transform appears when separating the real andimaginary parts of the complex quantities.


Represented in Fig. 3.1(b) for n = 5, it appears as a single circular rim around the origin, which peaksfor |α|2 = n, an intuitive result.

An interesting example of non-classical state is given by the ‘phase cat states’, which are superpo-sition of two coherent fields with opposite phases:∣∣∣Ψ±cat

⟩=

1√N±

(|β〉 ± |−β〉) , (3.223)

where:

N± = 2(1± e−2|β|2

), (3.224)

is a normalization factor resulting from the finite overlap of |β〉 and |−β〉. The density matrix is

ρ±cat =∣∣∣Ψ±cat

⟩⟨Ψ±cat

∣∣∣.Assuming again β real, and expanding coherent states scalar products, we get:

Q[cat,±](α) =1

πN±

[e−|α−β|

2+ e−|α+β|2 ± 2e−(|α|2+|β|2) cos(2βα′′)

]. (3.225)

The cat’s Husimi distributions are made up of two Gaussians [first two terms in the right-hand side ofeqn. (3.225)], corresponding to the field components and of an interference term, a Gaussian aroundthe origin multiplied by a modulated cosine function. The maximum amplitude of the interference

term, ∼ e−|β|2/π at α = 0, is exponentially small for large β values. The Q function of the

∣∣∣Ψ+cat

⟩cat state, plotted in Fig. 3.1(c) for n = 5, is thus almost identical to that of a statistical mixture,represented in Fig. 3.1(d). The Q function is not able to display the quantum coherence of such astate superposition. We thus introduce in the next paragraph the Wigner function that is much moresuited to the pictorial representation of the coherence of such cat states.

The Wigner function

The Wigner function is defined as the two-dimensional Fourier transform of the symmetric ordercharacteristic function:6

W (α) =1

π2

∫d2λCs(λ)eαλ

∗−α∗λ . (3.226)

Separating the real and imaginary parts and setting α = α′ + iα′′ = x+ ip, we can also write:

W (x, p) =1

π2

∫d2λCs(λ)e2i(pλ′−xλ′′) . (3.227)

The Wigner function is real, as a direct consequence of the conjugate property of its Fouriertransform (eqn. 3.200). It is also normalized, which results from the identity:∫

d2αW (α) =1

π2

∫d2αd2λCs(λ)eαλ

∗−α∗λ =

∫d2λCs(λ)δ(λ) = Cs(0) = 1 , (3.228)

where we have used again the integral definition of the two-dimensional Dirac distribution. Beforeexploring further the properties of W , we will write it in two other equivalent and useful forms.

We first make explicit the trace operation in the definition of Cs(λ), by using the continuouseigenbasis |x〉 of the quadrature operator X0 = (a+ a†)/2:

Cs(λ) =

∫dx′

⟨x′∣∣ ρD(λ)

∣∣x′⟩ . (3.229)

6To simplify the notation, we omit for the time being, since no confusion is possible, the indication of the state inthe expression of the Cs function and its Fourier transform. We write here Cs(λ) instead of C

[ρ]s (λ) and W (α) instead

of W [ρ](α). We come back later to the complete notation.


Using the Glauber identity, we can write:

D(λ) = e−iλ′λ′′e2iλ′′X0e−2iλ′P0 , (3.230)

It follows that:D(λ)

∣∣x′⟩ = eiλ′λ′′e2iλ′′x′

∣∣x′ + λ′⟩. (3.231)

Hence:

W (α) = W (x, p) =1

π2

∫dλ′dλ′′dx′ e2i(pλ′−xλ′′)eiλ

′λ′′e2iλ′′x′ ⟨x′∣∣ ρ ∣∣x′ + λ′⟩. (3.232)

The integral over λ′′ is a one-dimensional Dirac function:∫dλ′′ eiλ

′′(λ′+2x′−2x) = 2πδ(λ′ + 2x′ − 2x) . (3.233)

The integration over λ′ is then simple, leading to:

W (x, p) =2

π

∫dx′ e4ip(x−x′) ⟨x′∣∣ ρ ∣∣2x− x′⟩ . (3.234)

Setting finally u = 2(x′ − x), we obtain an important form for the Wigner function:

W (x, p) =1

π

∫du e−2ipu 〈x+ u/2| ρ |x− u/2〉 , (3.235)

which is the Fourier transform of non-diagonal density matrix elements in the position eigenstatesbasis. The function W is clearly sensitive to quantum coherence in the field state. The Fouriertransform can be inverted yielding the matrix elements of ρ in terms of W as:

〈x+ u/2| ρ |x− u/2〉 =

∫dp e2ipuW (x, p) . (3.236)

This equation shows that the knowledge of W is equivalent to that of the density operator, and hencethat the Wigner function contains all information needed to compute the expectation value of anyobservable in the state of the system. We come back to this point below.

An even simpler expression for W can be derived from eqn. (3.235). We start from the translationrelation: ∣∣∣∣x− u

2

⟩= e−i(x−u)pD(x+ ip)

∣∣∣∣−u2⟩, (3.237)

[which follows directly from eqn. (3.231) in which we set x′ = −u/2 and λ = x + ip] and from theconjugate relation, in which we change u into −u:⟨

x+u

2

∣∣∣∣ =

⟨u

2

∣∣∣∣D(−x− ip)ei(x+u)p . (3.238)

We then replace |x− u/2〉 and 〈x+ u/2| by their expressions (3.237) and (3.238) in eqn. (3.235) andwe introduce the hermitian parity operator P which performs a symmetry around the phase spaceorigin according to:

P |x〉 = |−x〉 ; P |p〉 = |−p〉 . (3.239)

We finally get:

W (x, p) =1

π

∫du

⟨u

2

∣∣∣∣D(−α)ρD(α)P∣∣∣∣u2⟩

=2

πTr[D(−α)ρD(α)P] . (3.240)

The Wigner function is thus the average value of 2P/π in the state obtained by displacing the oscillatorin phase space by the amount −α, transforming its density operator according to ρ→ D(−α)ρD(α).


It is easy to show that the P operator is the photon number parity observable. The |n〉 Fock statewave functions in the |x〉 and |p〉 representations are a product of an even parity Gaussian multipliedby a Hermite polynomial which has the parity of n. Reversing the sign of x or p in these functionsthus amounts to a multiplication by (−1)n, so that we can write:

P = eiπa†a . (3.241)

The parity operator is clearly hermitian and unitary, with the Fock states as eigenvectors and eigen-values ±1:

P |n〉 = (−1)n |n〉 . (3.242)

Being the expectation value of an observable, W is thus a directly measurable quantity. The eigenvaluesof the parity operator being +1 and −1, its expectation lies between these values. It follows that theWigner function is bounded:

−2/π ≤W (α) ≤ 2/π . (3.243)

As noted above, the average value of any observable can be obtained from the knowledge of theWigner distribution. There is in fact a very simple recipe to express this average by a simple integral,provided the observable, considered as a function of a and a†, has been cast in the symmetric order.The relation (3.226) defining W can be written:

W (α) =1

π2Tr

[ρ

∫d2λ eλ

∗(α−a)−λ(α∗−a†)], (3.244)

in which the integral is reminiscent of the two-dimensional Dirac distribution. We can then formallywrite:

W (α) = Tr [ρcδ(α− a)] . (3.245)

The Wigner function is thus the average value of the operator-Dirac distribution of α− a. Obviously,such a formal derivation has to be taken with care. We do not enter here in the mathematicaljustification of this expression, which is valid provided all quantum operators are put in the symmetricorder.

Consider now an observable O of the field mode. It can be expanded as a power series of a and a†

and cast in the symmetric order by repeated commutations of these operators. Let us note Os(a, a†)

the resulting expression. Formally, we can write:

Os(a, a†) =

∫d2α os(α, α

∗)δ(α− a) , (3.246)

where os(α, α∗) is the complex function of α obtained by replacing a by α and a† by α∗ in Os(a, a

†).The average value of Os is then:

〈Os〉 = Tr [ρOs] =

∫d2α os(α, α

∗)Tr [ρcδ(α− a)] =

∫d2α os(α, α

∗)W (α) . (3.247)

This simple expression reminds us of eqn. (3.195) giving the average value of any classical observableas an integral over the probability density in phase space. The real and normalized Wigner functionplays in this respect the role of f(x, p) in classical physics. We will see however that, contrary to f ,W can take negative values which are a signature of the oscillator’s quantum behaviour.

The analogy between f(x, p) and W (x, p) can be pushed further by considering the marginaldistributions of x and p, obtained by integrating the distribution over the conjugate variable. Settingu = 0 in eqn. (3.236), we get immediately the probability density of finding the value x of theoscillator’s position (or field quadrature X0) as:

P (x) = 〈x| ρ |x〉 =

∫dpW (x, p) . (3.248)


A simple calculation involving changes between the |x〉 and |p〉 bases shows that the integration of Wover x yields likewise the probability density of finding the value p of the oscillator’s momentum (orfield quadrature Xπ/2):

P (p) = 〈p| ρ |p〉 =

∫dxW (x, p) . (3.249)

More generally, the Wigner distribution can be expressed as a function of a any couple of orthogonalfield quadratures xφ and pφ defined by:

xφ = x cosφ+ p sinφ ; pφ = −x sinφ+ p cosφ , (3.250)

and corresponding to axes rotated in the phase plane by an angle φ with respect to the x and pcoordinate directions. The marginal distributions properties apply to any such couple of conjugatequadratures:

P (pφ) =

∫dxφW (xφ, pφ) . (3.251)

It can be shown that W is the only quantum phase space distribution obeying this interesting property,on which the ‘quantum tomographic’ methods for the reconstruction of W are based.

For many quantum states, the probability distributions P (x) or P (p) present nodes. This is, forinstance, the case for the x distribution of the excited Fock states. When P (x0) = 0, it follows that:∫

dpW (x0, p) = 0 . (3.252)

Hence, W cannot be everywhere positive. When it takes negative values, it cannot be assimilated toa genuine probability distribution. In fact, as we now show, negativities in W are clear-cut indicatorsof the non-classical nature of a field state. We now examine the Wigner functions of a few simplequantum states.

Let us consider first a coherent state |β〉 (including obviously the vacuum state). Using thesymmetric-order characteristic function [eqn. (3.207), in which we replace α by β], the correspondingWigner function, W [|β〉〈β|](α) is:

W [|β〉〈β|](α) =1

π2

∫d2λ e−|λ|

2/2eλ(β∗−α∗)−λ∗(β−α)

=2

πe−2|β−α|2 . (3.253)

It is a Gaussian, with a width 1/√

2, centred on the classical amplitude β and taking for α = β itsmaximum allowed value, 2/π. Figure 3.2(a) and (b) present the Wigner functions of the vacuum(β = 0) and of a coherent state with β =

√5 (n = 5 photons on the average).

The Wigner function of a thermal field, with nth photons on the average is derived also from thecorresponding Gaussian characteristic function (eqn. 3.213):

W [ρth](α) =2

π

1

2nth + 1e−2|α|2/(2nth+1) . (3.254)

It is again a Gaussian, centred at the origin and represented in Fig. 3.2(c) for nth = 1. The width isnow

√nth + 1/2 and the peak value is reduced to 1/π(nth + 1/2)

Let us examine also the case of a squeezed vacuum state, often considered in quantum optics as‘non-classical’. It is obtained by the action on the vacuum of the ‘squeezing’ operator:

S(ξ) = e(ξ∗a2−ξa†2)/2 , (3.255)

where we assume for the sake of this simple discussion that ξ is real. The squeezed states are minimaluncertainty states satisfying the Heisenberg uncertainty relation ∆X0∆P0 = 1/4. For a coherent


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0 . 6

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02

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02

4

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02

4

0

0 . 3

0 . 6

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02

4

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02

4

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02

4

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02

4

0

0 . 3

0 . 6

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02

4

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02

4

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02

4

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02

4

0

0 . 3

0 . 6

- 4- 2

02

4

- 4- 2

02

4( a ) ( b )

( d )( c )

Figure 3.2: Classical state Wigner functions. (a) Vacuum state. (b) Coherent state with β =√

5. (c) Thermal field

with nth = 1 photon on the average. (d) A squeezed vacuum state, with a squeezing parameter ξ = 0.5. The vertical

scales are from 0 to 2/π.

state, the variances of X0 and P0 are equal. For a squeezed state the standard deviation of the X0

quadrature is reduced to:

∆X0 =1

2e−ξ , (3.256)

and the fluctuations of P0 accordingly increased:

∆P0 =1

2eξ . (3.257)

Note that the limit of infinite squeezing corresponds to the position eigenstate |x = 0〉, with no positionfluctuations and infinite momentum uncertainty. The Wigner function of a squeezed vacuum is a non-degenerate Gaussian:

W [sq,ξ](x, p) =2

πe−2 exp(2ξ)x2

e−2 exp(−2ξ)p2, (3.258)

represented in Fig. 3.2(d) for ξ = 0.5. A general squeezed state corresponds to a complex ξ parameterand to a displacement towards a non-vanishing classical amplitude in phase space. Its Wigner functionis also a non-degenerate Gaussian, centred on the classical amplitude, whose principal axis (minimumand maximum fluctuations) are tilted with respect to x and p.

For all states considered so far, including the squeezed ones, the Wigner function is definite andpositive. It has all the properties of a classical probability distribution in phase space. For thecomputation of any observable eigenvalue, these states can be considered as classical fields with astochastic complex amplitude whose probability distribution over phase space is given by W . This isnot the case for the states we consider below.

The Fock state Wigner function can be derived from the corresponding characteristic function(eqn. 3.210):

W [|n〉〈n|](α) =2

π(−1)ne−2|α|2Ln(4|α|2) . (3.259)

Note that W [|n〉〈n|] does not depend upon the phase of α, which depicts the complete phase indeter-minacy in Fock states. Since Ln(0) = 1:

W [|n〉〈n|](0) =2

π(−1)n . (3.260)


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02

4

- 4- 2

02

4

- 0 . 3

0

- 4- 2

02

4

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02

4

Figure 3.3: Wigner function of a five-photon Fock state.

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02

4

- 20

2

- 0 . 5

0

0 . 5

- 0 . 5

0

0 . 5

- 4- 2

02

4

- 20

2

- 0 . 5

0

0 . 5

- 0 . 5

0

0 . 5

( a ) ( b )x x

p p

W W

Figure 3.4: Wigner functions of even (a) and odd (b) 10-photon π-phase cats

The Wigner function of odd photon number states takes the minimal value −2/π at the origin ofphase space. It cannot thus be interpreted as a probability distribution of classical field fluctuations,as was the case for the states considered above. In fact, all photon number states, but |0〉, presentnegativities in some region of phase space. The single-photon Wigner function:

W [|1〉〈1|](α) = − 2

π(1− 4|α|2)e−2|α|2 , (3.261)

has a Mexican-hat shape. As another example, we present in Fig. 3.3 the five-photon state Wignerfunction. It presents circular rims around the origin, with alternating negative and positive values.Note that this representation does not coincide with the intuitive picture of a Fock state, betterdepicted by the Husimi-Q function. The inner rims in W [|n〉〈n|] correspond to quantum interferencefeatures, revealing the non-classical nature of the Fock states.

Finally, a cat state Wigner function is:

W [cat,±] =1

π2N±

∫d2λ e(αλ∗−α∗λ)

[〈β|D(λ) |β〉+ 〈−β|D(λ) |−β〉

3.5. COUPLING FIELD MODES: THE BEAMSPLITTER 99

±〈β|D(λ) |−β〉 ± 〈−β|D(λ) |β〉]. (3.262)

The first line in this equation corresponds to the weighted sum of the Wigner functions of the coherentstates |β〉 and |−β〉. The second line corresponds to non-diagonal terms involving the quantumcoherence between the two components. These terms are absent in the case of a statistical mixture(|β〉〈β|+ |β〉〈β|)/2. Assuming, for the sake of simplicity, that β is real we get:

〈−β|D(λ) |β〉 = eiβλ′′ 〈−β |β + λ〉 = e−2β(β+λ)−|λ|2/2 . (3.263)

The coherence terms also correspond to Gaussian integrations which can be performed explicitly,leading finally to:

W [cat,±](α) =1

π(1± e−2|β|2)

[e−2|α−β|2 + e−2|α+β|2 ± 2e−2|α|2 cos(4α′′β)

]. (3.264)

The cat Wigner function is represented in Fig. 3.4(a). It exhibit large negative values near the phase-space origin, a clear signature of non-classical behaviour of a cat, depicted in a more vivid way thatthrough the Q distribution. Fig. 3.4(b) presents the Wigner function of an ‘odd’ cat, |β〉− |−β〉. Theonly difference is a π shift of the phase of the interference pattern.

We can discuss in more quantitative terms why is it that W is better adapted to the cat statesthan Q. The W and Q functions are connected in reciprocal space by the relation (3.204) betweentheir Fourier transforms, Cs(λ) being obtained from Can(λ) by a multiplication by e|λ|

2/2. We haveseen that this coherence is described by vanishingly small terms in Q around the phase space origin.The value of Q at α = 0 is equal to the integral over the λ plane of its Fourier transform Can(λ). Thisintegral is thus very small for a cat. The same integral, performed on the Fourier transform of W ,yields a much larger result since Cs(λ)/Can(λ) diverges for |λ| → ∞, entailing W (0) Q(0). Thefeatures of the phase space distribution around α = 0, critical to the description of the coherence, arethus greatly amplified by the e|λ|

2/2 multiplication transforming Q into W in reciprocal space.

As a final remark, we have defined W and Q as the Fourier transforms of Cs and Can. What aboutCn? Its Fourier transform is yet another phase space density function, the Glauber–Sudarshan-Pdistribution. It turns out to be highly singular. For instance, the P distribution of the coherent state|β〉 is a Dirac distribution centred at β and the P distribution of an |n〉 Fock state involves the nfirst derivatives of δ. In spite of their mathematical interest, these distributions do not provide a verypractical representation of quantum states.7

3.5 Coupling field modes: the beamsplitter

We now explore the quantum dynamics of coupled field modes, providing an useful insight into thephysics of beam-splitters, which are essential tools in quantum optics experiments8. We present asimple model of the beam-splitter, a device coupling linearly two modes of the field and examine thetransformations of a few simple imput state in this device.

3.5.1 Hamiltonian approach

Two possible realizations of a beamsplitter are shown in Fig. 3.5. On the left, two modes (a) and(b) of the radiation field with the same frequency propagate at right angles. At their intersection,there is a partially transmitting, partially reflecting dielectric slab inclined at 45 degrees with respect

7One can in fact consider a wide range of quasi-probability distributions, by defining the s-order of operators (s beinga continuous parameter), whose symmetric, normal and anti-normal orders are limiting cases. None of these ‘exotic’distributions is as useful as W or Q.

8Once again, this Section is derived from Haroche and Raimond, Exploring the quantum, OUP, 2006.


(A) (B)

( )a

( )a

( )a

( )a

( )b

( )b

( )b

( )b

Figure 3.5: Two beam-splitter models. (A) Two freely propagating modes coupled by a semi-transparent mirror. (B)

Two modes guided in single-mode fibres coupled in a fibre coupler. Incoming wave packets (dotted lines) and outgoing

ones (solid lines) are represented pictorially in both cases.

( )b( )a

j

-j

Figure 3.6: Principle of a generalized beam-splitter with adjustable phase ϕ. A semi-reflecting plate, inclined at 45

with the (a) and (b) beams, is sandwiched between two retarding plates inducing opposite phase shifts on beam (a).

to the beam directions. We assume that the modes are geometrically matched: they have the samepolarization (orthogonal to the incidence plane) and the spatial structure of mode (b) reflected bythe slab exactly matches the transmitted mode (a). Figure 3.5(b) represents an optical fibre couplerrealizing the same mode mixing. The upper fibre, (a), and the lower fibre, (b), are placed side by sideover some length,9 so that light is coupled from one mode into the other.

In both cases, the two field-mode amplitudes are classically mixed. In the case of a lossless dielectricslab, for instance, the emerging electric field amplitudes, E′a and E′b, are linked to the incoming ones,Ea and Eb, by a unitary and symmetrical10 matrix relation:(

E′aE′b

)= Uc

(EaEb

)=

(t(ω) r(ω)r(ω) t(ω)

)(EaEb

), (3.265)

where t(ω) and r(ω) are complex transmission and reflection coefficients, functions of the frequency ω ofthe field modes, which satisfy the unitarity conditions |t(ω)|2+|r(ω)|2 = 1 and t(ω)r∗(ω)+t∗(ω)r(ω) =0. The general relation (3.265) holds for any kind of lossless symmetrical semi-reflecting device,whether dielectric or metallic, and for a fibre coupler as well. By choosing properly the phase originof the transmitted field, one can always assume that t(ω) is real positive, which entails that r(ω) ispurely imaginary. We then introduce a mixing angle θ characterizing the beam-splitter operation at

9In practical couplers, the cladding around the fibre core is partially removed. The modes’ evanescent waves aroundboth fibre cores are thus superposed and directly coupled.

10The unitarity is imposed by energy conservation, while the symmetry of the matrix reflects the invariance of opticslaws when propagation directions are reversed.


frequency ω, set t(ω) = cos θ/2, r(ω) = i sin θ/2, and for a monochromatic field Uc is:

Uc(θ) =

(cos(θ/2) i sin(θ/2)i sin(θ/2) cos(θ/2)

). (3.266)

This description of classical mode mixing is valid for monochromatic fields interacting continuouslywith the beam-splitter. In general, a realistic experiment involves time-limited wave packets propa-gating in modes (a) and (b) and ‘colliding’ on the beam-splitter during a time τ , as illustrated on Fig.3.5. These transient fields are developed over a collection of field modes spanning a small frequencyinterval ∆ω = 1/τ . To avoid complications arising from the spectral analysis of this time-dependentproblem, we take the limit in which ∆ω is very small and τ very long yet finite, so that we can definea ‘before’ and an ‘after’ for the collision. We assume that the time envelopes of the wave packetsare identical and thus overlap perfectly on the beam-splitter. We consider also that the r, t and ϕparameters do not vary over the narrow spectrum of the wave packets, so that we can define a singlematrix (3.266) or (??) describing the same mixing process for all the frequency components of thelight field.

What is the action of this beam-splitter on the quantum states and observables of the field? Wenow show that it is described by an interaction Hamiltonian Hab, which, at the classical limit of largenumbers of photons, has the same effect on the field operators as the transformation (??) on theclassical fields. This Hamiltonian acts transiently on the fields, coupling the two quantum oscillatorsduring the finite time τ corresponding to the simultaneous passage of the wave packets on the beam-splitter. Since it describes a linear interaction, Hab is made of two terms associated to single-photonexchange between the two modes:

Hab(t) = −hg(t)

2(ab† + a†b) , (3.267)

where a and b are the annihilation operators in the modes and g(t) is a slowly varying real functionof time, switching adiabatically the coupling on and off for the time interval τ around t = 0. In thefollowing, we consider the action of Hab alone. This implicitly assumes that we use an interactionrepresentation with respect to the Hamiltonians of the free modes.11

The operation of the beam-splitter can be analyzed either in the Schrodinger or Heisenberg pointof views. In the first case, the quantum states of both modes evolve due to the coupling HamiltonianHab. This approach is used in the next section to describe the evolution of some special quantumstates.

We focus here on the Heisenberg point of view, which lends itself to a direct comparison betweenthe classical and quantum descriptions. The quantum states are constant and the dynamical variablesare the field operators. In the interaction picture with respect to the free mode Hamiltonians, theseoperators are time-independent before and after the ‘collision’ with the beam-splitter. We can thusidentify them, for t −τ and t τ , with the a and b operators of the Schrodinger picture. Thebeam-splitter transforms the input operator a into the output operator a′:

a′ = U †aU , (3.268)

where U is the evolution operator:

U = e−(i/h)∫Hab(t) dt , (3.269)

the integral being carried over the whole duration τ of the coupling. The evolution operator can berewritten as:

U(θ) = e−iGθ/2 , (3.270)

11To simplify the notation, we will omit the tilde that we should use in an interaction representation.


where:

G = −(ab† + a†b) and θ =

∫g(t) dt . (3.271)

Note that we omit the θ and ϕ arguments in U when no confusion is possible. We now have:

a′ = U †aU = eiGθ/2ae−iGθ/2 = a+iθ

2[G, a]

+i2θ2

2!22[G, [G, a]] + · · ·+ inθn

n!2n[G, [G, [· · ·, [G, a]]]] + · · · , (3.272)

where we have used again the Baker–Hausdorff lemma to develop the exponentials. The nestedcommutators on the right-hand side can be easily computed: [G, a] = b and [G, [G, a]] = a. All theodd-order terms in the development are proportional to b, while the even-order ones are proportionalto a. The coefficients of a and ib thus coincide with the series expansion of cos θ/2 and sin θ/2respectively:

a′ = U †aU = cos(θ/2) a+ i sin(θ/2) b , (3.273)

and similarly:b′ = U †bU = i sin(θ/2) a+ cos(θ/2) b . (3.274)

The output mode operators are a linear (unitary) combination of the input ones 12. These relations areformally identical to the input–output matrix relations (3.266) for the classical electric field amplitudes.This identity, which justifies the expression (3.267) of Hab, expresses the correspondence principle.When the fields involve many photons, the annihilation operators behave as the classical field complexamplitudes. The mixing parameter θ is directly related to the transmission and reflection coefficientsof the beam-splitter, which can be measured in a classical optics experiment. These results areindependent of the precise details of the model.

Taking the hermitian conjugate of the previous relations and noting that U †(θ) = U(−θ), we findsimilarly:

Ua†U † = cos(θ/2) a† + i sin(θ/2) b† ; Ub†U † = i sin(θ/2) a† + cos(θ/2) b† . (3.275)

3.5.2 State transformations

We now analyse the action of the beam-splitter on selected initial states in the Schrodinger point ofview. The final state |Ψ′〉 is given in terms of the input state |Ψ〉 by |Ψ′〉 = U |Ψ〉.

No photons

The first trivial case corresponds to the vacuum impinging in both modes, |Ψ〉 = |0, 0〉. This state isinvariant under the action of U . A beam-splitter has no effect in the dark.

One photon

Let us consider now a single photon impinging in mode (a), the initial state being |1, 0〉 [in thefollowing, the first and second labels in the kets refer to (a) and (b) respectively]. The sequence ofequalities:

U |1, 0〉 = Ua† |0, 0〉 = Ua†U †U |0, 0〉 = Ua†U † |0, 0〉 , (3.276)

leads, using eqn. (3.275), to:

U |1, 0〉 =[cos(θ/2) a† + i sin(θ/2) b†

]|0, 0〉 = cos(θ/2) |1, 0〉+ i sin(θ/2) |0, 1〉 . (3.277)

12We should by the way obtain similar expressions in a mode basis change involving only two modes. This paragraphon the beamsplitter operation is thus also appropriate for the description of field state modifications in a mode basischange.


For a single photon impinging on the beam-splitter in mode (b), we get:

U |0, 1〉 =[i sin(θ/2) a† + cos(θ/2) b†

]|0, 0〉 = i sin(θ/2) |1, 0〉+ cos(θ/2) |0, 1〉 . (3.278)

A single photon is thus in general dispatched in a coherent superposition between the two modes,with a final probability distribution corresponding to the classical light intensity transmission andreflection coefficients cos2(θ/2) and sin2(θ/2). In particular, for θ = π/2, we have:

U(π/2) |1, 0〉 = (|1, 0〉+ i |0, 1〉)/√

2 . (3.279)

Is this an entangled state? If we define the system as two oscillators (a) and (b) coupled by thebeam-splitter, the answer is definitely ‘yes’. The two parts of this bipartite entity are entangled bysharing in an ambiguous way a quantum of excitation. Correlations between measurements performedindependently on the two modes could reveal this entanglement. If we renounce to perform suchindependent measurements, though, we can define two new modes (±), linear combinations of (a) and(b) corresponding to the photon creation operators (a† ± ib†)/

√2. The state (3.279) then represents

a single photon in the (+) mode with the vacuum in the (−) mode and, in this point of view, it isnot entangled. This example shows that the notion of entanglement depends upon the basis in whichwe choose to describe the system (this choice being in turn determined by the kind of measurementswe intend to perform on it). The unitary transformation changing the tensor products of Fock statesin modes (a) and (b) into those of photon number states in modes (±) is non-local with respect tothe (a)-(b) partition since it mixes the two modes. Such a non-local operation can result in differentdegrees of entanglement in the two representations, turning an entangled state into a separate one.13

Adding, by means of a mere dephaser in one of the modes, an adjustable phase ϕ between thetwo-components of the single-photon state (3.277), we can explore, when θ and ϕ are tuned, the wholetwo-dimensional Hilbert space spanned by |1, 0〉 and |0, 1〉. We note the formal analogy between asingle photon ‘suspended’ between the two modes and a spin on the Bloch sphere. It has been suggestedto exploit this analogy and to code information into the path of single-photon bimodal states, |1, 0〉and |0, 1〉, playing the roles of logical |0〉 and |1〉 qubit states. In this context, the beam-splitter whichtransforms the logical qubit states into their most general superpositions realizes a one-qubit quantumgate.

n photons

Let us now consider the action of the beam-splitter on fields containing many photons. Assume firstthat n-photons impinge in (a), the initial state being |n, 0〉. A calculation similar to the single-photonone yields:

U |n, 0〉 = U(a†)n√n!|0, 0〉 =

1√n!U(a†)nU †U |0, 0〉 . (3.280)

Noting that U(a†)nU † = (Ua†U †)n, and using again eqn. (3.275):

U |n, 0〉 =1√n!

[cos

θ

2a† + i sin

θ

2b†]n|0, 0〉 . (3.281)

The right-hand side can be expanded according to the binomial law:

U |n, 0〉 =n∑p=0

(n

p

)1/2

[cos(θ/2)]n−p [i sin(θ/2)]p |n− p, p〉 . (3.282)

13Note that the notion of entanglement is here independent of the number of particles in the system. A single photoncan be in an entangled state or not, depending upon the basis choice.


The final quantum state is a coherent superposition of terms corresponding to all possible partitionsof the n incoming photons between the two modes. If we describe the system as a bipartite (a)-(b)entity, the final state is thus in general massively entangled. When θ is an even multiple of π, theoutput state is disentangled, in fact identical to the initial one [all photons in mode (a)]. When θ isan odd multiple of π, all photons are channelled in mode (b) (again a disentangled situation).

To be specific, let us consider in more detail a balanced beam-splitter (θ = π/2). The output stateis:

U (π/2, 0) |n, 0〉 =1√2n

n∑p=0

(n

p

)1/2

(i)p |n− p, p〉 . (3.283)

The probability of getting n − p photons in (a) and p in (b) is proportional to(np

), the number of

partitions of the incoming photons into the two outputs. From the photon counting point of view, thequantum beam-splitter behaves as a random device. Each photon ‘freely chooses’ to emerge either in(a) or (b), with equal probabilities. The average photon number in each output mode is thus n/2. Thephoton number in one mode obeys, in the limit of large incoming photon numbers n 1, Gaussianstatistics with a width

√n/2. The relative fluctuations, proportional to 1/

√n, are negligible for large

fields and the two output energies are balanced according to the correspondence principle. This simplepicture, treating photons as billiard balls, is valid because we consider only the output energies. Itis not sufficient to account for the massive entanglement between the two modes, which would berevealed by more complex photon coincidence experiments.

Coherent states

Suppose now that the impinging field in mode (a) is in a coherent state |α〉 with n = |α|2 photons onthe average, while mode (b) is in vacuum. The output state of the beam-splitter is now:

U |α, 0〉 = UDa(α)U † |0, 0〉 , (3.284)

where Da(α) is the displacement operator in mode (a). Using the operator identity Uf(A)U † =f(UAU †), which holds whenever f can be expanded in power series, we get:

UD(α)U † = eαUa†U†−α∗UaU† . (3.285)

Using eqn. (3.275) and its hermitian conjugate, and separating the commuting contributions of a andb in the exponentials, we get:

U |α, 0〉 = Da [α cos(θ/2)]Db [iα sin(θ/2)] |0, 0〉 , (3.286)

where Db is the displacement operator acting on mode b. The output state is thus:

U |α, 0〉 = |α cos(θ/2), iα sin(θ/2)〉 . (3.287)

It is an unentangled tensor product of coherent states in the two modes, with amplitudes obeyingthe same relations as the classical fields (eqn. 3.266). The coherent state dynamics, entirely classicaleven for small photon numbers, is thus very different from that of Fock states. The non-entanglementof coherent states involved in linear coupling operations is a very basic feature of these states, withfar-reaching consequences, as we will see below when discussing relaxation and decoherence.

Let us note also that the two modes exchange periodically their energies as a function of themixing angle θ or, equivalently, as a function of the coupling time. This periodic energy exchange isthe counterpart of the classical beating phenomenon for two coupled mechanical oscillators.

3.6. FIELD RELAXATION 105

Photon collision on a beamsplitter

A beam-splitter can also be used to mix two excited modes. As a simple example, suppose that (a)and (b) initially contain a single photon. The output state is then:

U |1, 1〉 = Ua†b† |0, 0〉 = Ua†U †Ub†U † |0, 0〉 . (3.288)

Using eqn. (3.275), we get:

U |1, 1〉 =i sin θ√

2[|2, 0〉+ |0, 2〉] + cos θ |1, 1〉 , (3.289)

which is, in general, an entangled state. In the case of a balanced beam-splitter (θ = π/2):

U(π/2, 0) |1, 1〉 = (|2, 0〉+ |0, 2〉) /√

2 , (3.290)

which is a superposition of two photons propagating together in mode (a) or (b). The two photonsare bunched in the same mode after the beam-splitter. This can be seen as a consequence of thebosonic nature of the photons. It is also a quantum interference effect. The probability amplitudeof getting one photon in each mode cancels, because of the exact destructive interference betweentwo indistinguishable quantum paths. In one of the paths, both photons are transmitted, with aprobability amplitude t × t =1/2. In the other, they are both reflected, with the quantum amplitudeir × ir = −1/2.

This process and the resulting state are non-classical. A superposition of two photons either inmode (a) or in mode (b) is the simplest example of a multi-particle quantum state superposition.This photon ‘bunching’ effect has been observed by the first time by Hong, Ou and Mandel in theearly times of quantum optics. It is generally considered as one of the key phenomena exhibiting fieldquantization.

It is obtained here because the polarizations of the two photons are the same. Photons withorthogonal polarizations behave differently. If they are initially in an entangled polarization state,anti-symmetric versus photon exchange, they emerge ‘anti-bunched’ in different spatial modes. Thisis required to preserve the overall state symmetry versus photon exchange. Photon entanglement andBell states analysis techniques are based on this feature. Let us add that these photon coincidenceexperiments are technically difficult, since the wave packets must be well-matched, both in space(mode matching) and in time (coherence length overlap).

3.6 Field relaxation

We now treat the important problem of the coupling of a mode, not to a single other one as in thebeamsplitter case, but to a continuum of modes playing the role of an environment. We describe inthis way for instance the quantum field state transformation due to the attenuation in an optical fibre.The diffusion on the silica defects, after a long path, couples photons to the outside. We also describewith this situation the important case of Cavity Quantum Electrodynamics, when the box limitingthe field is not a virtual one, but a real cavity made up of mirrors. The residual losses or diffusion onthese mirrors also induce a damping of the quantum states of the field.

3.6.1 Lindblad equations

The system S (in the terms used to discuss relaxation in Chapter 2) is a field mode at frequency ω,coupled to the environment E . What are the possible jump operators? Assuming, reasonably, thatthe coupling with the environment is linear in the cavity field amplitude, there are only two possiblejumps. The mode may lose a photon in the environment or gain a photon from it.


The environment simulator B has, accordingly, three different states, the initial one,∣∣∣0(B)

⟩,∣∣∣−(B)

⟩and

∣∣∣+(B)⟩

, corresponding respectively to the loss or to the gain of a photon by the cavity field.

These latter states are associated to the jump operators L− and L+. The first, L−, is proportionalto the annihilation operator a: L− =

√κ−a. The second, L+, is proportional to the photon creation

operator: L+ =√κ+a

†.

We can relate the rates κ− and κ+ by a general thermodynamical argument, independent of thespecifics of E . Any process in which S loses (or gains) a quantum must be accompanied by a correlatedjump of the environment in the opposite direction, a transition upwards (or downwards) on the energyladder of E . The probabilities for these transitions to occur are proportional to those for finding in Ean initial state linked to a final state having with it an energy difference ±hωc, necessary to conservethe energy of the S+E system. A given transition in the environment will thus contribute to the gainand loss of photons in a ratio equal to the probability of finding E in the upper state of this transitiondivided by the probability of finding it in its lower state. Assuming that the environment is in thermalequilibrium at temperature T , this ratio is simply equal to exp(−hω/kbT ) (Boltzmann law). We thushave:

κ+ = κ−e−hω/kbT . (3.291)

The relation between the loss and gain rates can be expressed in a more transparent form, byintroducing the average number nth of thermal photons per mode at frequency ωc, given by Planck’slaw:

nth =1

ehω/kbT − 1. (3.292)

Equation (3.291) can then be rewritten as:

κ−κ+

=1 + nth

nth, (3.293)

which leads us to define κ− and κ+ in term of a unique damping rate κ as:

κ− = κ(1 + nth) ; κ+ = κnth , (3.294)

and to write the Lindblad equation for the field state ρ as:

dρ

dt= −iωc

[a†a, ρ

]− κ(1 + nth)

2

(a†aρ+ ρa†a− 2aρa†

)−κnth

2

(aa†ρ+ ρaa† − 2a†ρa

). (3.295)

The commutator term in this equation can be removed by switching to an interaction representationwith respect to the field Hamiltonian, that we chose here as H ′0 = hωa†a (we get rid of the vacuumenergy term). The relaxation terms are not modified, since a is changed into a exp(−iωt) while a†

becomes a† exp(iωt).

3.6.2 Evolution of some states

Fock states

The field master equation turns into a simple rate equation for the photon number distribution p(n) =ρnn = 〈n| ρ |n〉. Using eqn. (3.295) and the action of a and a† on Fock states, we obtain a closed setof equations for p(n):

dp(n)

dt= κ(1 + nth)(n+ 1)p(n+ 1) + κnthnp(n− 1)− [κ(1 + nth)n+ κnth(n+ 1)]p(n) . (3.296)


The first two terms on the right-hand side describe the transitions towards the n-photons state |n〉,increasing its occupation probability. They originate either from the upper state |n+ 1〉, at a rateκ(1 + nth)(n + 1) or from the lower level |n− 1〉, with a rate κnthn. The last term describes thetransition out of state |n〉, towards |n+ 1〉 or |n− 1〉. At zero temperature, the damping rate of theFock state |n〉 is nκ.

These rates can be used to determine the photon statistics in the steady-state solution of themaster equation. We invoke a ‘detailed balance argument’, stating that the transfer rates from |n〉 to|n− 1〉 and from |n− 1〉 to |n〉 are equal when the system A ceases to evolve:

κ(1 + nth)np(n) = κnthnp(n− 1) , (3.297)

which leads immediately to:p(n)

p(n− 1)=

nth

1 + nth= e−hω/kbT . (3.298)

We thus find, not unexpectedly, that the mode S ends up in the Boltzmann equilibrium distributionat the temperature T . The average steady-state photon number is then given by Planck’s law (3.292).

It is also instructive to determine how the average photon number 〈N〉 =∑n np(n) reaches this

equilibrium. Calculating the rate d 〈N〉 /dt =∑n ndp(n)/dt with the help of eqn. (3.296) leads to:

d 〈N〉dt

= −κ(〈N〉 − nth) . (3.299)

The κ coefficient is the exponential decay rate of the field mean energy towards its thermal equilibrium,a quantity which is classically denoted as ω/Q, where Q is the dimensionless mode quality factor.In spite of the obvious environment’s complexity, mode relaxation depends only upon two simpledimensionless parameters, Q and nth. It is interesting to note that the damping rate of the Fock state|n〉 is n times larger than that of the energy in the mode. This is a typical decoherence effect. Asillustrated by the negativities in their Wigner functions, the Fock states are non-classical. The largerthey are, the more fragile with respect to the coupling with the environment.

It is very instructive to apply the quantum Monte Carlo approach to mode relaxation. We startby considering the evolution from an initial Fock state |n〉. Consider first an environment at zerotemperature. The single jump operator, L− =

√κa, describes the loss of a photon by the cavity

mode. The non-hermitian part of the Hamiltonian describing the ‘no-jump’ evolution reduces to

He = −ihJ = −ihκa†a/2 . (3.300)

The effective Hamiltonian is proportional to the mode one H ′0 = hωa†a. The no-jump evolution isthus formally obtained by setting the mode frequency at the complex value ω − iκ/2.

During the first time interval τ , the probability of a jump is p1 = τκ 〈n| a†a |n〉 = κτn, an intuitiveresult, which could have been guessed from eqn. (3.296). Being an eigenvector of H ′0, |n〉 is alsoan eigenvector of the non-hermitian ‘no-jump’ Hamiltonian. Besides an irrelevant phase factor, thestate thus does not evolve in the no-jump case. When a jump occurs, |n〉 is transformed into |n− 1〉.Throughout the evolution, the mode thus remains in a Fock state, the photon number decreasing byone unit at each jump. The field energy is a staircase function of time. The jump rate is the largestat the start, when the photon number is maximum. The last photon is lost at a rate κ, which is alsothe classical energy damping rate.

Figure 3.7 presents simulations of a 10-photon Fock state decay. The staircase curves correspondto random individual quantum trajectories. All eventually end up in the vacuum state |0〉. Theevolution of the standard deviation, ∆N , of the photon number can be guessed from these curves.Zero in the initial and final states, it reaches a maximum during the evolution. The solid line presentsthe evolution of the average photon number obtained from 10 000 such trajectories. It is in agreement


0 1 2 3 4 50

2

4

6

8

1 0

n

t / Tc

Figure 3.7: Quantum Monte Carlo simulation of a 10-photon Fock state relaxation in a T = 0 K environment. Photon

number n as a function of time in units of the mode damping time Tc = Q/ω. Thin lines (solid, dashed and dotted):

three individual staircase trajectories. Thick line: average over 10 000 trajectories corresponding to the usual exponential

decay.

with the exponential decay with a rate κ predicted by the density matrix approach, which cannot bedistinguished from the statistical average at this scale.

For a finite environment temperature T , L− is multiplied by√

1 + nth and another jump operatorL+ =

√nthκa

† must be considered. The two possible jumps lower or increase the photon numberby one. The field is again at any time in a Fock state, but there is no stationary final state for anytrajectory. After a transient regime lasting a time ∼ 1/κ during which the average field energy adjuststo its equilibrium value, the photon number keeps jumping up and down randomly around nth.

Coherent states

Let us now describe the T = 0 K Monte Carlo trajectory starting from an initial coherent state |β〉.The properties of this trajectory, which may seem paradoxical at first sight, will explain the importantfeatures of coherent state relaxation: no entanglement with the environment and exponential decayof the field amplitude.

The probability for counting a photon in the first time interval is p1 = κτ〈β|a†a |β〉 = κτ |β|2 = κτn.If a photon is counted, the state is unchanged since |β〉 is an eigenstate of the jump operator L− =

√κa.

An evolution occurs, on the contrary, if no photon is recorded in this time bin. The imaginary frequencycontribution to the effective Hamiltonian produces a decrease of the state complex amplitude, the fieldevolving, in the interaction picture with respect to its free Hamiltonian, according to:

|β〉 →∣∣∣βe−κτ/2⟩ . (3.301)

This situation is quite paradoxical, since the field’s state amplitude decreases only if no photonis recorded! How is this possible? First of all, how comes that the state does not change when weare sure that one photon has been lost? We have here the combination of two effects, which tend tochange the photon number in opposite directions. On the one hand, the loss of a quantum reducesthe average energy in the cavity. On the other hand, the registration of a photon click modifies ourknowledge about the state in which the field was just before the photon was emitted, correspondingto an increase by precisely one unit of the photon number expectation at this time.

To understand this point, let us ask a question whose answer is a simple exercise in conditionalprobabilities: knowing the a priori Poissonian photon number distribution p(n) = e−nnn/n! (centredat n = n at time t) and that a photon has been lost and detected as a ‘click’ c in the time interval(t, t+ δt), what is the conditional probability p(n|c) that the field mode contained n photons at time


t? If the photon emission probability were n-independent, p(n|c) would merely be equal to p(n). Theprobability of losing a photon is however proportional to the expectation value of a†a, hence to n.Detecting a photon has thus the effect of shifting the probability distribution towards larger n′s, sincethe corresponding states are more likely to lose quanta.

This result can be made quantitative by invoking Bayes law on conditional probabilities. Theprobability we seek, p(n|c), is related to the reverse conditional probability p(c|n) of detecting a clickduring the same time interval, conditioned to the prior existence of n photons. This probability isobviously p(c|n) = κnδt. It is slightly different from the a priori probability pc = κnδt of detecting aclick when the field is in the coherent state, a superposition of n states around the mean n. We canthen relate p(n|c) and p(c|n) by expressing in two different ways the joint probability p(n, c) that aclick is detected during the considered time interval and that the field contains n photons:

p(n, c) = p(c|n)p(n) = p(n|c)pc , (3.302)

which yields immediately the Bayes law result:

p(n|c) = p(n)p(c|n)

pc=n

np(n) = e−n

nn−1

(n− 1)!= p(n− 1) . (3.303)

This formula shows that p(n|c) is precisely equal to the a priori probability that the initial field containsn− 1 photons. The maximum of p(n− 1), hence of p(n|c), occurs for n = n+ 1. The knowledge thata click has occurred biases the photon number distribution before this click towards larger n values,with a photon number expectation exceeding the a priori average value by exactly one unit. The lossof the photon signalled by the click brings this number back down to n. The two effects exactly canceleach other and the state of the field has not changed!

This is a very peculiar property of coherent states, a consequence of their Poissonian photonnumber distribution. For non-Poissonian fields with larger fluctuations, this effect could be even morecounter-intuitive and result in an overall increase of the photon expectation number, just after theloss of a photon! We leave the reader to construct a photon number distribution leading to such asituation. Note finally that this counter-intuitive increase of the field energy upon loss of photonsrequires the existence of fluctuations in the initial energy distribution. The effect does not occur fora Fock state which jumps down the energy ladder at each click and loses its energy in a perfectlyintuitive way.

We have now to understand the other side of the coherent state evolution paradox. Why are theydecreasing in amplitude precisely during the time intervals when no photon clicks are registered? Wehave already encountered the same effect (continuous quantum state evolution without jump) whenanalysing the spontaneous emission of a single atom in a superposition of its emitting and non-emittingstates. It is here again a matter of conditional probability. If no photon is detected in a time interval,it is more likely that the photon distribution has less photons that was a priori assumed.

To consider an extreme situation, a coherent field has a small probability, equal to e−n, to containno photon at all. This is the probability that the detector will never click, however long one waits.The longer the time interval t without click, the more likely it becomes that the field is effectively invacuum. Its wave function evolves continuously without jump under the effect of the non-hermitianoperator 1− iκa†a/2. If t becomes very long, the field ends up effectively in |0〉. This seems counter-intuitive, but we must not forget that this event has a very small probability of occurring. Thisexample is once again a striking illustration of an important aspect of measurement theory. Even theabsence of a click is information whose knowledge affects the wave function of the field.

In Monte Carlo simulations, the randomness is generated only by the jumps. The evolutionbetween the jumps is continuous and deterministic. The insensitivity of a coherent state to jumpsmakes its Monte Carlo trajectory certain. To determine the state at time t of a field initially in state|β〉, we have to piece together its partial evolutions during the successive intervals between jumps t1,


t2, . . . , ti, . . . , tN adding up to t. Whatever the distribution of these intervals, the final state is uniquely

defined, in the interaction picture, as∣∣∣βe−κ(t1+t2+....+tN )/2

⟩=∣∣∣βe−κt/2⟩. The field state remains pure,

as already indicated by eqn. (3.301).

Chapitre 4

Coupling quantum field to matter

This Chapter completes our program. We now have quantum atoms and a quantum description of thefield and we will couple these two systems. In a first Section of the Chapter, we will give the couplingHamiltonians of charges with a quantum field in some simple situations. We will study the case ofa classical macroscopic current and show that classical sources produce the most classical states, thecoherent states of the field. We shall then give the general Hamiltonian coupling a quantum atom tothe quantum field.

The next Section will be devoted to spontaneous emission. We will be able to compute from thefirst principles the spontaneous emission rate, through two complementary approaches. We will alsotouch the level shifts problems. The divergence of the Lamb shift is a clear indication that our simpleapproach to field quantization should be supplemented, for some problems, by the full renormalizationof QED. Nevertheless, our approach leads to interesting results in simple cases.

The third Section will be devoted to a model of a field detector, an essential component in anyquantum optics experiment. We will use this model to revisit some fundamental experiments, such asthe intensity correlations observed for the first time by Hanbury-Brown and Twiss.

The final Section of this Chapter will be devoted to an in-depth description of Cavity QuantumElectrodynamics. This very active domain pushes the matter-field coupling to its simplest limits,with a single atom coupled to a single field mode containing a few photons. All the difficulties of themulti-mode problems disappear. The dynamics, in spite of being highly non-trivial, can be explicitlycalculated and checked experimentally. The domain has provided a few key illustrations of quantumphysics that we will shortly describe.

4.1 Interaction Hamiltonians

4.1.1 Quantum field and classical charges

We start by a short description of the last semi-classical model in these lectures. Instead of consideringa quantum atom coupled to a classical field, we treat here the case of a quantum field coupled to aclassical oscillating current, made of so many elementary charges that all its quantum features can beneglected. We write this current:

j(r, t) = j0(r)e−iω0t , (4.1)

where the amplitude j0 may be complex.

We couple this current to all field modes. We will remove from the field Hamiltonian the constantenergy offset of the vacuum state (we remove here an infinite quantity out of the Hamiltonian, butthis is not a problem since we will treat separately all modes at the end).

The field Hamiltonian is thus

H ′0 =∑`

hω`a†`a` , (4.2)

111

112 CHAPITRE 4. COUPLING QUANTUM FIELD TO MATTER

where we use a prime to distinguish it from the Hamiltonian including the vacuum energy contribution.The classical interaction energy of the current with the field is the integral over all the space of

the scalar product −j ·A of the current with the vector potential. We will extend this and write theinteraction Hamiltonian Hi as

Hi = −∫V

j(r, t) ·A(r, t) d3r , (4.3)

where

A(r, t) =∑`

√h

2ε0ω`Va`f`(r) + c.c. . (4.4)

Note that all the dependence in r is in the classical parts of the vector potential. The interactionHamiltonian is thus a combination of a` and a†` operators in the field’s Hilbert space.

Before proceeding further, we will get rid of the free evolution of the modes by switching to aninteraction representation with respect to H ′0. The state vector is then:∣∣∣Ψ⟩ = U †0 |Ψ〉 , (4.5)

withU0 = e−iH

′0t/h , (4.6)

which factors in independent operators on each modes:

U0 =∏`

e−iω`ta†à` (4.7)

The new Hamiltonian isH = U †0HiU0 . (4.8)

To get the Hamiltonian in the interaction representation, we need first to transform a`, a simpletask since all modes transform independently:

a` = eiω`ta†ààè

−iω`ta†à`

= a` − iω`ta` +(iω`t)

2

2a` + . . .

= aè−iω`t , (4.9)

where we have used the Baker Hausdorff Lemma and the fact that [N`, a`] = −a`.We inject that in the interaction Hamiltonian and get

H = −∫d3r

[j0(r)e−iω0t + j∗0(r)eiω0t

]·[∑

`

√h

2ε0ω`V

(aè−iω`tf`(r) + a†è

iω`tf∗` (r))]

. (4.10)

Obviously, we consider here only positive frequencies ω` and ω0. The action of the current on themode will be notable only if ω` ≈ ω0. We can thus neglect all modes which are not close of the currentfrequency. For the remaining ones, we can again perform a variant of the rotating-wave approximation(RWA) by keeping in H only the terms evolving at slow frequencies and discarding those evolving closeto 2ω0. With this approximation and performing the integrals, we get finally:

H = −∑`

√hV

2ε0ω`J0a†è−i(ω0−ω`)t + h.c. , (4.11)

where the complex scalar J0 is defined as:

J0 =1

V

∫d3r j0(r) · f∗` (r) . (4.12)

4.1. INTERACTION HAMILTONIANS 113

Note that we have defined J0 so that it has the dimension of a current density (f` is dimensionless).Since H is a sum of independent Hamiltonians for each mode, we can treat the dynamics of each

mode separately. We will thus follow a single one and drop the mode index `. Setting

K0 =

√hV

2ε0ωJ0 (4.13)

we can write the Hamiltonian in the simpler form

H = −K0e−iδta† + h.c. , (4.14)

whereδ = ω0 − ω , (4.15)

is the detuning between the source and the mode.We now compute the evolution under this Hamiltonian, starting from the initial vacuum state. It

is important to note that the Hs taken at different times are non-commuting operators. The evolutionoperator is thus not simply the exponential of the integral of the Hamiltonian.

To cope with this problem, we consider the elementary evolution operator from time t to timet+ dt. During this short time interval, the Hamiltonian can be taken as:

H = −K0e−iΦa† + h.c. , (4.16)

whereΦ = δt . (4.17)

The elementary evolution operator U(t, dt) is the exponential of this Hamiltonian, and is clearly ofthe form exp(αa† − α∗a), a displacement operator with the infinitesimal complex amplitude:

α =iK0

he−iΦdt . (4.18)

The total evolution operator is the product of all these elementary displacements. The productof two displacements is, within a phase, the displacement corresponding to the total displacement.Within a global phase (that we leave to the reader as an exercise), the final state at time t is thus acoherent state with an amplitude β:

β =

∫ t

0

iK0

he−iδt

′dt′ = −K0

hδ

[e−iδt − 1

]. (4.19)

When δ 6= 0, the amplitude β propagates in phase space along a circle, a simple result. The fieldamplitude periodically goes back to zero, an reaches a maximum amplitude for δt = kπ, where kis an odd integer. This periodic excitation and de-excitation of the field mode is a typical beatingphenomenon.

The situation is different when δ = 0. In this case, the amplitude β evolves linearly with timeaccording to:

β =iK0

ht . (4.20)

The number of photons in the mode, |β|2 evolves quadratically. This runaway process is a directconsequence of the bosonic nature of the photons. The more photons in the field, the easier to addsome new ones.

We have thus shown in this paragraph that a classical source produces a coherent state in eachof the modes it is coupled to. These states are thus rather easily produced, be it with electronicoscillators in the radio-frequency domain or with lasers in the optical domain.


4.1.2 Quantum field and quantized atom

Writing the interaction Hamiltonian of a quantized atom with a quantized field is quite easy. We justreplace in the different forms of the semi-classical Hamiltonian (obtained in the dipole approximationregime and neglecting the non-linear terms in the field) the fields or potentials by their classicalcounterparts.

We will use either

Hap = − q

mP ·A(0) , (4.21)

or

Hde = −D ·E(0) . (4.22)

More specifically,

D = dεd |g〉〈e|+ h.c. , (4.23)

where d is the (possibly complex) dipole matrix element and εd a unit vector describing the polarizationof the atomic transition. The electric field at the origin (where the atom is assumed to be located) is

E(0) = i∑`

√hω`2ε0V

a`ε` + h.c. , (4.24)

where we have used here for instance the plane wave mode expansion.

Developing the D ·E scalar product, we find four types of terms. Two of them involve the operatorσ−a

† and its conjugate. They describe processes in which the atom gets excited (de-excited) whileabsorbing (emitting) a single photon. The two other types of terms involve σ+a

† and its conjugate.They correspond thus to counter-intuitive processes in which the atom gets excited while emitting aphoton.

When the atom-field coupling is not too large, and for modes whose frequency is close to that ofthe atomic transitions, the two latter terms can be safely neglected. This is yet another variant of theRotating Wave Approximation.

4.2 Spontaneous emission in free space

We apply first these Hamiltonians to the calculation of the spontaneous emission rate. We considera single two-level atom initially in its upper state |e〉. The atom, coupled to the set of modes of alarge (virtual) quantization box emits a photon when jumping in the ground state |g〉 (the |e〉 → |g〉transition is at frequency ω0).

We will use two complementary approaches to derive the spontaneous emission rate; The first,semi-qualitative one, makes use of a Fermi Golden Rule expression. The second, more detailed andpredictive, reproduces the first calculations of Wigner and Weisskopf. It also puts in evidence theunderlying difficulty of the diverging light shifts.

4.2.1 Fermi Golden Rule

We use the electric dipole form of the Hamiltonian. The field is expanded on the plane wave modes, ina very large quantization box. The initial state is |e, 0〉 (atom in the upper state and no photon), thefinal state is obviously |g, 1`〉 (atom in the ground state with one photon in the mode indexed by `. Wehave a discrete initial state coupled by the Hde Hamiltonian to a continuum of final states. The FermiGolden Rule tells us that the evolution is irreversible, exponential, and proceeds at a rate proportionalto the square of the coupling Hamiltonian matrix element times the density of final states.

The orientation of the outgoing photons will be important to compute the scalar products of theatomic and field polarization in the Hamiltonian. We must thus treat separately all final photon

4.2. SPONTANEOUS EMISSION IN FREE SPACE 115

directions. We assume that the quantization box is so large that there are still infinitely many modeswith a wavevector pointing in a small solid angle dΩ. Obviously, in the Fermi Golden Rule situation,the emission in separate sets of modes just add independently. We will thus add up the emission ratesin all solid angles dΩ. We should take care that, in dΩ, all modes have nearly the same directionof propagation, but have two different possible polarizations (linear polarizations if we take linearlypolarized mode basis).

We can thus write the spontaneous emission rate Γ as

Γ =

∫dΓdΩ , (4.25)

where dΓ is the rate corresponding to an emission in dΩ. It is given by the Fermi Golden Rule:

dΓ =∑ε`

2π

h|W |2dρ(E = hω0, dΩ) , (4.26)

where W is the coupling Hamiltonian matrix element and dρ the density of states in the solid angle dΩat the atomic transition energy (the density of sates should be in energy terms for the Fermi Goldenrule) and the sum runs over the two possible polarizations in this solid angle.

Let us first evaluate the density of states dρ. An isotropy argument leads immediately to dρ =ρdΩ/4π, where ρ is the total density of states per unit energy. We have already computed, in the firstSection of the previous Chapter, the density of states in units of frequency1

ρ(ν) =8π

2c3Vν2dν . (4.27)

Writing ρ(E)dE = ρ(ν)dν for E = hν = hω0, we arrive at

ρ(E) =V

2π2c3

1

h

(E

h

)2

, (4.28)

and hence finally at

dρ(E = hω0, dΩ) =V

8π3

ω20

hc3dΩ . (4.29)

Let us now evaluate the coupling term:

|W |2 = | 〈g, 1`|D ·E |e, 0〉 |2 . (4.30)

In order to simplify a bit the algebra, and without real loss of generality, we will set in this expression:

εd = uz , (4.31)

i.e. we will use a linearly polarized atomic transition. Keeping only the RWA terms, the only term inthe interaction Hamiltonian coupling the two states is that involving a and σ+. Hence

|W |2 =

∣∣∣∣∣duz · ε∗`√hω`2ε0V

∣∣∣∣∣2

. (4.32)

We can now fully evaluate dΓ and we find, using ω` = ω0:

dΓ =∑ε`

1

8π2ε0

ω30

c3

|d|2

h|uz · ε∗` |2dΩ . (4.33)

1Note that we use here a total density of states, not a volumic density (hence the factor V. Moreover, we have a factorof two less than in the Planck’s law calculation, since we will perform the sum over the two orthogonal polarizationslater.


Calling ε1 and ε2 the two linear (hence real) orthogonal polarizations, we can expand uz on the basisof these vectors, together with the unit vector in the direction of propagation, uk. Since all thesevectors are unit vectors, we get

(uz · ε∗1)2 + (uz · ε∗2)2 = 1− (uz · uk)2 = 1− cos2 θ = sin2 θ , (4.34)

where the solid angle dΩ is defined by the spherical coordinates angles θ and φ.

We can now write the total emission rate by integrating over all solid angles:

Γ =1

8π2ε0

ω30

c3

|d|2

h

∫ 2π

0

∫ π

0sin3 θ dθdφ (4.35)

The angular intergral is easy and simplifies to 8π/3. We thus get the very important result, alreadyannounced a few times in these lecture notes:

Γ =ω3

0|d|2

3πω0hc3. (4.36)

4.2.2 Wigner-Weisskopf

We now give the more detailed Wigner-Weisskopf approach. It starts of course from the same hypothe-ses, but does not make use of the Fermi Golden rule. Instead, we shall write the explicit dynamics ofthe atom-field system in the Schrodinger picture. At time t, the complete state can be expanded as

|Ψ(t)〉 = c0(t) |e, 0〉+∑`

c`(t) |g, 1`〉 . (4.37)

Letting

V` = −〈e, 0|D ·E |g, 1`〉 , (4.38)

we can put the Schrodinger equation for |Ψ〉 under the form of an infinite set of coupled lineardifferential equations:

ihdc0

dt= hω0c0 +

∑`

V`c` (4.39)

ihdc`dt

= hω`c` + V ∗` c0 , (4.40)

with the initial conditions c0 = 1, c` = 0. We can formally integrate the second equation. Setting

b` = cèiω`t , (4.41)

we get the equation for b` as

ihdb`dt

= eiω`tV ∗` c0 , (4.42)

and hence

b`(t) =V ∗ìh

∫ t

0c0(t′)eiω`t

′dt′ (4.43)

or

c`(t) =V ∗ìh

∫ t

0c0(t′)eiω`(t

′−t) dt′ (4.44)

We can inject this formal solution in the equation for c0 and get rid of the simple term −iω0c0 bysetting

c0 = e−iω0tα0(t) . (4.45)

4.2. SPONTANEOUS EMISSION IN FREE SPACE 117

We getdα0

dt= −

∑`

|V`|2

h2 eiω0t∫ t

0eiω`(t

′−t)e−iω0t′α0 dt′ . (4.46)

Changing for the variable τ = t− t′, we get

dα0

dt= −

∫ t

0N (τ)α(t− τ) dτ , (4.47)

where the integral kernel N is:

N (τ) =1

h2

∑`

|V`|2ei(ω0−ω`)τ . (4.48)

Using now the expression of the coupling Hamiltonian, we can make V` explicit as in the previousparagraph and get:

N (τ) =|d[2

h2

[∑`

|uz · ε`|2hω`2ε0V

e−iω`τ]eiω0τ . (4.49)

It is quite clear from this expression than N (τ) decays very rapidly versus τ . The sum over allthe modes involves terms of similar orders of magnitude and vastly different frequencies, which veryrapidly come out of phase with each other. In a time of the order of 1/ω0, N practically vanishes.For all practical purposes, it can be considered as being proportional to a Dirac distribution. This isthe Markov approximation, which holds as long as the reservoir has a wide enough spectrum to havebasically a zero memory time.

Thus: ∫ t

0N (τ)α(t− τ) dτ ≈ α0(t)

∫ ∞0N (τ) dτ =

(Γ

2+ i∆

)α0(t) . (4.50)

The equation of evolution of α0 (remember than α−0 is, within a phase, identical to c0 and thus that|α0|2 is the probability for finding the system in its initial state) is thus:

dα0

dt= −

(Γ

2+ i∆

)α0 . (4.51)

In this equation, Γ clearly appears as the spontaneous emission rate and ∆ as a frequency shift of theatomic transition.

The solution of this equation is obvious and, returning to the original coefficients, we finally find

c0(t) = e−Γt/2e−iω0te−i∆t , (4.52)

and, injecting that in the formal solution of the equation for c`:

c`(t) =V`ih

1− e−Γt/2ei(ω`−ω0−∆)t

(Γ/2)− i(ω` − ω0 −∆). (4.53)

Taking the square modulus of this coefficient in the limit of large times:

|c`|2 =|V`|2

h2

1

(Γ2/4) + (ω` − ω0 −∆)2, (4.54)

we get the final probability for having a photon in mode ` i.e. the spectrum of the spontaneousemission. Not surprisingly, we find a Lorentzian spectrum with a width Γ centred at the atomicfrequency ω0 shifted by ∆.


The last step to complete the calculation is thus to integrate the kernel N and to find its real andimaginary parts. We get:(

Γ

2+ i∆

)=|d[2

h2

∑`

(uz · ε`)2 hω`2ε0V

∫ ∞0

ei(ω0−ω`)τ dτ . (4.55)

Using the distribution identity ∫ ∞0

eiωt dt = πδ(t) + iPP 1

ω, (4.56)

we can separate easily the Γ and ∆ parts.

For Γ , we get

Γ =2π|d|2

h2

∑`

(uz · ε`)2 hω`2ε0V

δ(ω0 − ω`) , (4.57)

which is exactly the expression of the Fermi golden rule if we replace the sum by an integral over adensity of modes. We leave it to the reader to check the exact identity with the result of the previousparagraph.

What about ∆ now? We will not enter into the details, but the calculation merely diverges, duein particular to the contribution of the infinite frequency modes. The atomic transitions should beshifted by an infinite amount, a quite non-physical result.

This problem plagued the early days of quantum electrodynamics. The solution cam rather late,during the Shelter island congress of 1947. Lamb had experimentally observed the slight shift betweenthe energies of the 2S and 2P levels. This shift results from the coupling to the vacuum modes.Computing it was thus a really important problem. It implied the development of renormalizationtheory.

We will not treat it here of course. Suffice it to say that it provides a consistent way to computefinite energy level shifts. We can nevertheless continue our exploration of the atom-field coupling.Most interesting phenomena, be it only the spontaneous emission, rely on the coupling with a fewnearly resonant modes. For these processes, the infinitely many high frequency modes play no role.We can thus simply take into account the effect of these modes as (measurable) shifts of the energylevels and compute the finite effect of the resonant modes without any mathematical problem.

4.3 Photodetection

Photodetection is a key ingredient in quantum optics. After all, any quantum optics experimentfinally ends up in detecting photons. We propose in this Section a simple model of a photodetectorand apply it to some interesting quantum experiments, such as the Hanbury-Brown and Twiss intensitycorrelations.

4.3.1 Photodetector model

We need to model a broadband photodetector, such as a photomultiplier or a photodiode. It is quiteclear that a two-level atom will not do. We thus consider instead a system with a ground state |g〉and a continuum of excited states |ei〉, coupled to the ground state by electric dipole transitions. Weconsider that the detector has ‘clicked’ when it has made a transition from its lower state to any of itsexcited states. Note that our model, though simple, is a rather good description of a detector basedon photo-ionisation.

The detector Hamiltonian is:

Hd =∑i

hωi |ei〉〈ei| , (4.58)

4.3. PHOTODETECTION 119

where ωi is the transition frequency from |g〉 to |ei〉. The detector-field coupling Hi is the −D · EHamiltonian, where, with a simple generalization of the two-level model:

D =∑i

di(εi |g〉〈ei|+ ε∗i |ei〉〈g|) . (4.59)

We are interested here in the qualitative behaviour of the model. We will thus neglect most of thefactors and keep only the essentially important terms. We can thus rewrite the interaction Hamiltonianas:

Hi =∑i

κi |ei〉〈g|E+ + h.c. , (4.60)

where E+ is the positive frequency part of the proper polarization component of the electric field.The factor κi encapsulates the dipole matrix elements and all numerical factors. We assume here, forthe sake of simplicity, that the detector system is located at the origin.

We switch now to an interaction representation with respect to the detector and field free Hamil-tonians. This interaction representation amounts simply into replacing |ei〉〈g| by exp(iωit) |ei〉〈g| andall the a` operators in E+ by a` exp(−iω`t), making E+ a time-dependent operator. Finally, theHamiltonian reduces in this interaction representation to

Hi =∑i

κieiωit |ei〉〈g|E+(t) + h.c. , (4.61)

We now write the Schrodinger equation for the sate vector of the detector+field system and solveit with the initial condition

|Ψ(0)〉 = |g〉 ⊗ |Ψf 〉 , (4.62)

where |Ψf 〉 is the field state to detect. The formal solution of the evolution equation is

|Ψ(t)〉 = |g〉 ⊗ |Ψf 〉+1

ih

∫ t

0Hi(t

′)∣∣Ψ(t′)

⟩dt′ . (4.63)

We cannot solve this integro-differential equation, but we can find a perturbative solution at thefirst order in the field-detector coupling. We assume that the time is short enough so that the detectoris still mainly in state |g〉 and replace, in the integral, |Ψ(t′)〉 by |Ψ(0)〉 = |g〉 ⊗ |Ψf 〉.

Noting furthermore that, in Hi, the |g〉〈ei|E− part gives zero when applied to the initial state, wehave

|Ψ(t)〉 = |g〉 ⊗ |Ψf 〉+1

ih

∑i

κi

[∫ t

0dt′ eiωit

′E+(t′) |Ψf 〉

]⊗ |ei〉 . (4.64)

The probability pe for having observed a ‘count’ at time t is simply:

pe =∑i

| 〈ei |Ψ〉 |2 =∑i

〈Ψ |ei〉〈ei |Ψ〉 . (4.65)

Injecting in this expression the first-order solution of the Schrodinger equation, we get:

pe =1

h2

∑i

|κi|2∫ t

0dt′

∫ t′

0dt′′ eiωi(t

′−t′′) 〈Ψf |E−(t′′)E+(t′) |Ψf 〉 . (4.66)

For a high enough density of states (this is certainly the case for any practical detector) the sum overthe final states can be replaced by an integral with the introduction of a density of states:

∑i

−→∫dωρ(ω) . (4.67)


The relevant levels for the detector are nearly resonant with the incoming field. In this narrow spectralregion, we can furthermore assume that the coupling κ, a priori a function of ω is nearly constant.

The double integral giving pe contains then an integral over the final state frequency:∫dωeiω(t′−t′′) = πδ(t′ − t′′) . (4.68)

Only one time integral is left and we finally arrive at

pe(t) ∝∫ t

0dt′ 〈Ψf |E−(t′)E+(t′) |Ψf 〉 . (4.69)

If we consider now a detector made of very many such systems (located at the same point ofcourse), the ‘intensity’ of the signal, I, is proportional to dpe/dt. Hence, the ‘photocurrent’ is:

I(t) = 〈Ψf |E−(t)E+(t) |Ψf 〉 . (4.70)

The detected signal is proportional at any time to the average value of the E−E+ term in the fieldstate, a very simple result, quite compatible with the expectations from classical photodetection theory(intensity proportional to the square modulus of the field). Note also that the field operator productis here in the normal order (involving the photon number operator a†a), so that the vacuum fieldcorresponds to a zero detected intensity.

4.3.2 Intensity correlations

The interference effects are not limited to the classical (or quantum) addition of the field amplitudes.They can also be directly evidenced on the intensity of the photodetection signal itself, even if thefields are treated in the classical formalism. The realization of this fact by Hanbury-Brown and Twisshas first provided an interesting method for high resolution astronomy, but is has also played a keyrole in the development of quantum optics, since the intensity correlation signal directly reveals thestatistical properties of the quantum state.

Classical Hanbury-Brown and Twiss experiments

We shall first consider a simpler version of the HBT experiment which provides without calculationsan insight into the intensity correlations already in the classical field domain.

The classical light produced by a source is split in two equal parts by a beamsplitter and sent totwo detectors recording the intensity. The source may be for instance a spectral lamp emitting trainsof exponentially damped field pulses, with a correlation time τc. A system records the intensities I1(t)and I2(t) and computes their correlation function:

G2(τ) = I1(t)I2(t+ τ) , (4.71)

where the overline describes an average over a long time period or over very many realizations. Thecorrelation function G2 is independent of t if we make the hypothesis of a stationary phenomenon. Ifwe think, for a short while, in terms of photons, G2 is proportional to the probability for detectinga photon at t + τ knowing that a photon has been detected at t. The (for the time being) classicalcorrelation function gives thus a detailed insight into the field statistical properties.

For a very large time difference τ , the signals recorded by the two detectors are independent, sincethe source has a short memory time τc. Hence,

G2(∞) = (I)2 , (4.72)

where I is half of the mean intensity sent by the source.


D1

D2

I1

I2

D1 d

L

r1A

D2

A B

r1B

R

(a) (b)

Figure 4.1: (a) An autocorrelation experiment for a simple source. (b) Geometry of the HBT experiment for the

determination of stellar diameters.

At short times, and particularly for τ = 0, the two recorded intensities are not independent. Indeed

G2(0) = I2 , (4.73)

the average of the square intensity. Since

I2 − (I)2 = (I − I)2 ≥ 0 , (4.74)

it follows that, for any classical source:

G2(0) ≥ G2(∞) . (4.75)

The intensity autocorrelation function is a decreasing function of time. This can be interpretedsimply. When the source has an intensity fluctuation above the average (simultaneous emission of twopulses for instance), this surge is equally split between the two detectors, recorded by both of them.The correlation is thus maximal for short times and the width of the maximum of the correlationfunction is clearly of the order of τc.

HBT in their Nature paper (178, 1046) had a different purpose. They wanted to measure theangular diameters of stars too small to be resolved by conventional astronomy. They thus installedtwo small telescopes at a distance d apart, pointing at the same star at a huge distance L (that wewill assume to be at the zenith) (figure 4.1). They measured the intensities correlation function for azero delay τ as a function of d. Why is it that it contains information about the star size? For that,we need to make a more precise calculation, yet in the classical framework.

Let us consider two emitting points on the extremities of a diameter of the star, A and B. Sincethe star is a quite classical source, these two points emit totally uncorrelated fields, with randominstantaneous phases.

The envelope field received by D1 is thus of the form:

E1 = αeikr1AeiφA + βeikr1BeiφB , (4.76)


where r1A and r1B are the distances of the detector with the two emitting points, k is the wavevectorof light and φA and φB are the time-dependent random phases of the field pulses emitted on bothsides, which have an extremely short mutual correlation time τc. The factors α and β encapsulate allthe geometrical and dimensional factors. Note that, since the phases are random,

I1 = |E1|2 = |α|2 + |β|2 . (4.77)

A similar expression holds for E2 of course

E2 = αeikr2AeiφA + βeikr2BeiφB . (4.78)

Let us now compute the measured intensity correlation function at zero delay, I1(t)I2(t). By ex-pressing it in terms of the instantaneous electric fields, replacing the fields by their detailed expressionand expanding all squares and products, we do find a rather complex expression.

However, since we average on time, we should keep only those terms which do not involve a randomphase. The computation is painful, but one gets at the end

I1(t)I2(t) = I1 I2 + |α|2|β|2 cos k [r1A + r2B − r1B − r2A] . (4.79)

For a symmetric configuration, when A and B are symmetric with respect to the mediating segmentD1 −D2, r1A − r2A = r2B − r1B and

I1(t)I2(t) = I1 I2 + |α|2|β|2 cos 2k(r1A − r2A) = I1 I2 + |α|2|β|2 cos 2ω

c(r1A − r2A) , (4.80)

where ω is the field pulsation.The zero-time intensity correlation signal thus varies simply (as a cosine) as a function of the

difference of the light’s flight time from A to detectors 1 and 2, a very simple result indeed. CallingR the star radius, Θ its angular diameter (see figure 4.1), we get

I1(t)I2(t) = I1 I2 + |α|2|β|2 cos kdΘ . (4.81)

A measurement of the correlation function versus d thus provides a direct measurement of the star’sdiameter. The resolution is excellent and equivalent to the diffraction limit of an ideal telescope witha diameter d. In fact, the intensity correlation directly measures the difference between the time offlight of light from the star to the two detectors. The method is thus quite impervious to atmosphericperturbations. The only condition is that the time of flight perturbations induced by the atmosphericturbulence are less than the source correlation time. This is always the case, even at a low altitude.

Quantum intensity correlations

Let us now interpret the intensity correlation experiments from the quantum side. The single photode-tector current is the average value of the product E−E+. We extend that to the correlation functionsof the intensity received at time t in point r1 with the intensity received at time t + τ at point r2.Without further justification, let us admit that this correlation function writes

G2(r1, r2, t, τ) = 〈Ψf | G2 |Ψf 〉 , (4.82)

where we definedG2 = E−(r1, t)E

−(r2, t+ τ)E+(r2, t+ τ)E+(r1, t) . (4.83)

If we take τ = 0 and r1 = r2, it is clear for a single mode situation that G2 is proportional to (a†)2a2.We shall also define a normalized correlation function by

g2 =G2

I1 I2, (4.84)

where the denominator is made up of the product of the average intensities in r1 and r2.


Two field modes

Let us consider first the simple situation of two field modes, labelled 1 and 2 impinging at once ontwo detectors DA and DB located in rA and rB

2. This is the quantum version of the classical HBTexperiment. The two modes correspond to the positive frequency fields (projected over the commonpolarization direction of the two detectors:

E+i = Eiei(k1·r−ωit)ai , (4.85)

where the index i stands for 1 or 2. The total positive frequency component of the electric field is thus

E+ = E+1 + E+

2 . (4.86)

We will assume that the field state is a tensor product of Fock states in each mode, |Ψ〉 = |N1, N2〉.We will consider two photo-detection signals: the simple photo-current detected at one of the

detectors:

G1(rA, t) = 〈Ψ|E−(rA, t)E+(rA, t) |Ψ〉 , (4.87)

and the two-point two-times correlation G2(rA, tA, rB, tB) defined above.Let us first show that there can be no interference effect in the single-point single-time detection.

Expanding in the expression of G1 the fields in their components, we find four terms. Due to theorthogonality of the Fock states, these terms give a non-vanishing contribution to G1 only when theyconserve the photon number in each mode. The only surviving terms are proportional to a†1a1 and a†2a2.They give a contribution proportional to N1E2

1 and N2E22 respectively, and all phase factors cancel out.

The total detection probability is thus merely the sum of the intensities of the two modes and thereare no interference effect (this is due to the fact that the field state is a product of photon-numberstates, which contain no phase information).

Let us now write explicitly the G2 correlation function. Once again, the only remaining terms inthe expansion of the field operators must conserve the number of photons in each mode. Only fourterms survive (with their obvious hermitian conjugates), with field operators proportional to a†1a

†1a1a1,

a†2a†2a2a2, a†1a

†2a1a2 and a†2a

†1a2a1. The first two terms have no phase factors and are proportional to

Ni(Ni − 1)E4i . Only the two last terms can contribute to an interference effect. Indeed,

〈Ψ| a†1a†2a1a2 |Ψ〉+ h.c. = 2E2

1E22N1N2Re

[ei[(k2−k1)·(rA−rB)−(ω2−ω1)(tA−tB))]

]. (4.88)

The other term has a similar expression. This term is obviously an interference one, sensitive to thephase difference of the relative mode phases at the two points of interest. One can check easily thatthe phase differences here are identical to those involved in the classical HBT experiment discussion.In fact, any quantum interference results from the coherent addition of the probability amplitudeslinked to two indistinguishable quantum paths. Here, they correspond to one photon of mode 1 beingdetected in A while a photon of 2 is detected in B on the one hand, and one photon of 1 detected inB and one photon of 2 detected in A on the other hand. From an experimenter’s perspective, thosetwo paths cannot be distinguished if the frequency of the photons is not measured, and they interfere.See the Cohen-Tannoudji-Guery-Odelin for an in-depth discussion.

Single emitter: anti-bunching

We now consider the spontaneous emission by a single atom and compute the two points and twotimes field correlation function. We will not give a fully detailed calculation, but only a qualitativediscussion, which, nevertheless, captures all the essential phenomena. The experiment we have inmind is thus a single atom placed at the source location in the elementary HBT experiment.

2This discussion is derived from the Cohen-Tannoudji, Guery-Odelin book.


Let us thus compute the correlation:

G2(r, 0, r, τ) ≡ G2(τ) , (4.89)

which is clearly proportional to the probability for detecting a photon in r at time τ knowing that aphoton has already been detected at the same point at time zero. Of course, this function would beidentically zero if the atom is not excited by an external source. If this is not the case, it will emit asingle photon, and there is no room for a second emission. We thus assume that there is some coherentpumping mechanism (continuous laser or trains of laser pulses) which repeatedly re-excites the atom.

We will use for this semi-qualitative computation an Heisenberg picture in which the operatorsdepend upon time, the state being constant. We thus need the field operator E+(τ) as a functionof time. This field is produced by the atom through the D · E electric dipole coupling term. In theclassical realm, the field radiated is proportional to the atomic dipole. In the quantum realm, we willthus admit that the positive frequency electric field operator E+, which freely evolves as exp(−iωt)in the Heisenberg picture is proportional to the atomic dipole (which evolves at a similar frequency)or, more simply, to the operator σ−(τ). Hence, the correlation is proportional to

G2(τ) = 〈σ+(0)σ+(τ)σ−(τ)σ−(0)〉 (4.90)

At the origin of time, σ−(0) = |g〉〈e| and σ+(0) = |e〉〈g|. At time τ , σ±(τ) = U †σ±(0)U , where Uis the atomic evolution operator describing the pumping process. We can thus write

G2 =⟨|e〉〈g| [U † |e〉〈g|U ][U † |g〉〈e|U ] |g〉〈e|

⟩. (4.91)

We must here evaluate the average value in the initial atomic state, which is clearly |e〉(since a photonwill be detected at time zero, the atom must be excited at that time). We thus have:

G2(τ) = | 〈e|U(τ) |g〉 |2 , (4.92)

a quite simple result (of course, computing U might be a more difficult problem, but we do not enterin it.

For τ = 0, U = 11 and G2 = 0. We get here the main result of this discussion. When a single atom isused as emitter, the field correlation at zero time is also zero. This is quite different from the classicalHBT case, in which the correlation was always maximal at short times. The interpretation of thephenomenon is trivial. Once the atom has emitted a photon, it is in the ground state and cannot emitanother photon before the pumping process (described by the evolution operator U) has promoted itagain into |e〉. The photons emitted by an individual quantum emitter are thus ‘anti-bunched’ whereasthe photons emitted by a classical source tend to be ‘bunched’ and to exhibit positive correlation atshort times.

For longer times τ the evolution leads back to level |e〉 and the probability for observing the secondphoton raises. The correlation function raises and reaches, perhaps after some damped oscillationsreflecting the Rabi oscillations of the atom in the pumping field, the asymptotic value of uncorrelatedevents.

The observation of this effect by Kimble and Mandel in 1977 (PRL 39, 691) is, together withthe Hong Ou Mandel photon bunching on a beamsplitter one of the founding experiments of modernquantum optics. The anti-bunching effect is still widely used to assess the single-emitter nature of aquantum source.

4.4 The dressed atom model

We now examine an important situation, that of a single two-level atom strongly coupled to a singlefield mode. This ideal situation describes quite well the Cavity Quantum Electrodynamics (CQED)

4.4. THE DRESSED ATOM MODEL 125

experiments, as explained in the next Section. It is also a fair description of the coupling of an atomto a strong single-mode laser field, so strong that the coherent atom-field coupling overwhelms thedissipative processes, due for instance to spontaneous emission from the atomic exited levels.

We will see in this Section that this simple problem can be approached by an explicit, exactdiagonalization of the atom-field Hamiltonian, introducing the so-called ‘dressed states’. Proposed inthe 60s by Claude Cohen-Tannoudji, they have been widely used since then. They are a mandatorystep to treat the Cavity Quantum electrodynamics situation, in which the field mode only contains afew photons. They are also quite useful in the large field limit, since they lead to a simpler and moreinsightful solution than the Optical Bloch equations.

4.4.1 The atom-field eigenstates

Hamiltonian

The atom-field Hamiltonian simply reads:

H = Ha +H ′c +Hac , (4.93)

where Ha and H ′c = hωcN are the atom and field Hamiltonians.3

Performing the rotating-wave approximation, the atom–cavity coupling term reduces to:

Hac = −ihΩ0

2

[aσ+ − a†σ−

], (4.94)

where we introduce the ‘vacuum Rabi frequency’ Ω0:

Ω0 = 2dE0ε

∗d · εch

, (4.95)

where εc is the field mode polarization unit vector. We assume, for the sake of simplicity, that ε∗d · εcis real and positive, hence Ω0.4 The frequency Ω0 measures the strength of the atom–field coupling.It is proportional to the interaction energy of the atomic dipole with a classical field having the r.m.s.amplitude of the vacuum.

Let us examine briefly the ‘uncoupled’ eigenstates of Ha +H ′c. They are the tensor products |e, n〉and |g, n〉 of atomic and field energy states. Their energies are h(ωeg/2 + nωc) and h(−ωeg/2 + nωc).When the atom–field detuning:

∆c = ωeg − ωc , (4.96)

is equal to 0 or small compared to ωc, the uncoupled states |e, n〉 and |g, n+ 1〉 are degenerate ornearly degenerate. The excited states of Ha+H ′c are thus organized as a ladder of doublets, separatedfrom each other by the energy hωc (Fig. 4.2). The ground state |g, 0〉, representing the atom in itslower state in the cavity vacuum, is the only unpaired state at the bottom of this ladder.

The nth excited doublet stores n+1 elementary excitations either as n+1 field quanta (|g, n+ 1〉)or as the sum of n photons added to one atomic excitation (|e, n〉). The operator M = a†a + σ+σ−,representing the total number of atomic and field excitations, commutes with H and is a constant ofmotion. The excitation number being conserved, the atom–field coupling Haf connects only statesinside each doublet. The diagonalization of the full Hamiltonian, discussed in the next paragraph thusamounts to solving separate two-level problems.

3We take the vacuum as the zero energy for the field, getting rid of the hωc/2 offset. The field quantities are indexedwith a c in anticipation of the Section on cavity quantum electrodynamics.

4If this is not the case, we can always make Ω0 real positive by multiplying it by a proper phase term.


Eigenenergies and eigenvectors

Let us call Hn the restriction of the total Jaynes–Cummings Hamiltonian to the nth doublet. Intro-ducing the ‘n-photon Rabi frequency’:

Ωn = Ω0

√n+ 1 , (4.97)

we can write Hn in matrix form as:

Hn = hωc (n+ 1/2) 11 + Vn , (4.98)

with:

Vn =h

2

(∆c −iΩn

iΩn −∆c

)=h

2[∆cσZ + ΩnσY ] . (4.99)

The diagonalization of Hn is thus formally identical to the determination of the eigenstates of aspin placed in a magnetic field whose components along OZ and OY are proportional to ∆c and Ωn.This fictitious field makes with Z the angle θn defined by:

tan θn = Ωn/∆c . (4.100)

This ‘mixing angle’, varying between 0 and π, is useful to express in a simple form the eigenstates ofthe atom–field system, which are represented in the |e, n〉 , |g, n+ 1〉 basis by the same amplitudesas the |0, 1〉θn,ϕ spin states whose Bloch vectors point along the θn, ϕ = π/2 direction on the Blochsphere. From this simple analogy, we deduce immediately that the eigenvalues of Hn are:

E±n = (n+ 1/2) hωc ±h

2

√∆2c + Ω2

n , (4.101)

with the corresponding eigenvectors:

|+, n〉 = cos(θn/2) |e, n〉+ i sin(θn/2) |g, n+ 1〉 ;

|−, n〉 = sin(θn/2) |e, n〉 − i cos(θn/2) |g, n+ 1〉 . (4.102)

These states, which are generally entangled, are called the ‘dressed states’ of the atom–field system.The variation of the dressed energies as a function of ∆c is represented in Fig. 4.3. For very large

detunings, the dressed energies almost coincide with the uncoupled state energies (n+1/2)hωc±h∆c/2.At zero detuning, the uncoupled energy levels cross. The atom–cavity coupling transforms this crossinginto an avoided crossing, the minimum distance between the dressed states being the coupling energyhΩn. We examine now in more detail two interesting limiting cases: the resonant case (∆c = 0) andthe ‘large detuning case’ (∆c Ωn).

4.4.2 Resonant case: Rabi oscillation induced by n photons

In the resonant case, the mixing angle is θn = π/2 for all n values. The dressed states, separated byhΩn, are:

|±, n〉 = [|e, n〉 ± i |g, n+ 1〉] /√

2 . (4.103)

We have here a situation similar to two coupled degenerate oscillators, whose eigenmodes are the sym-metric and anti-symmetric superpositions of the independent free modes. The eigenmode frequencydifference corresponds then to the beating note in the correlated motion of the two oscillators. Here,the separation of the dressed states at resonance corresponds to the frequency of the reversible energyexchange between the atom and the field.

To be more specific, let us consider the simple case of an atom initially (t = 0) in state |e〉, inside acavity containing n photons. The initial state, |Ψe(0)〉 = |e, n〉, expanded on the dressed states basisis:

|Ψe(0)〉 = [|+, n〉+ |−, n〉] /√

2 . (4.104)


- 3 - 2 - 1 0 1 2 3 4

0

Energy

| e , n ñ

| g , n + 1 ñ

| + , n ñ

| - , n ñ

D c / W n

h W n

Figure 4.3: Dressed state energies as a function of the atom–cavity detuning ∆c. The uncoupled state energies are

represented as dotted lines.

We will describe the evolution of this state in the interaction representation with respect to theconstant term hωc(n+ 1/2)11 in eqn. (4.98). At time t, |Ψe(0)〉 becomes:∣∣∣Ψe(t)

⟩=[|+, n〉 e−iΩnt/2 + |−, n〉 eiΩnt/2

]/√

2 . (4.105)

The probabilities of finding the atom in |e〉 or |g〉 are obtained by reverting to the uncoupled basis:∣∣∣Ψe(t)⟩

= cosΩnt

2|e, n〉+ sin

Ωnt

2|g, n+ 1〉 . (4.106)

Similarly, if the atom is initially in state |g〉 in a cavity containing n + 1 photons,5 the combinedatom–field state at time t is:∣∣∣Ψg(t)

⟩= − sin

Ωnt

2|e, n〉+ cos

Ωnt

2|g, n+ 1〉 . (4.107)

The atom–field coupling results in a reversible energy exchange between |e, n〉 and |g, n+ 1〉 at fre-quency Ωn. This is the quantum version of the Rabi oscillation phenomenon which can be understoodas a ‘quantum beat’ between the two dressed states.

For the initial state |e, 0〉 (or |g, 1〉), this oscillation occurs at the frequency Ω0, hence the name ofvacuum Rabi frequency coined for this parameter. This situation will be described in details in thenext Section about CQED As the coupling of two field modes by a beam-splitter, the Rabi oscillationproduces in general an entanglement (eqns. 4.106 and 4.107).

For a mesoscopic coherent field, with an average photon number n of a few units, the Rabi oscil-lations signal involves a sum of components at the frequencies Ωn. The resulting signal is surprisinglycomplex, with a collapse of the oscillations due to the field amplitude dispersion, and a revival atlong times due to the quantization of the amplitude. These phenomena will be discussed in the nextSection also.

5Note that an atom in |g〉 does not evolve in the cavity vacuum since the state |g, 0〉 is an eigenstate of the fullatom–field Hamiltonian.


| e , n - 1 ñ

| g , n ñ

| e , n ñ

| g , n + 1 ñ

| + , n ñ

| - , n ñ

| + , n - 1 ñ

| - , n - 1 ñ

we g+ d w

e g wc+ d

ew

c

wc+ d

gw

c

Figure 4.4: Dressed state energies in the far-detuned case for two adjacent manifolds. The effect of the non-resonant

interaction is a slight shift of the atom–field energy levels. This results in atomic transition and field mode frequency

shifts, illustrated by the vertical arrows.

Let us examine the case of a large classical field, represented by a coherent state with a very largeaverage photon number n. The relative dispersion of the photon number, 1/

√n is the extremely small,

and all dressed atoms multiplicities have nearly equal splitting, Ωn = Ω0

√n (we can safely neglect 1

in front of n). All evolutions in all the nth doublets are exactly the same and we recover the classicalresult, a simple sinusoidal Rabi oscillation between |e〉 and |g〉 at the frequency Ωn. In other words,for all practical purposes and as far as only the Rabi oscillation is concerned, a large coherent statecan be assimilated to a Fock state.

Of course, even in a large Fock state, the Rabi oscillation would entangle the atom and the field(|n〉 and |n+ 1〉 are always orthogonal). This entanglement is an artefact of assimilating the coherentstate with a Fock state. Adding or subtracting a single photon to a large coherent state does notappreciably modify it. The field can thus be taken out of the dynamics evolution, and the dressedstates are reduced to two atomic superposition states, (|e〉 ± i |g〉)

√2, eigenstates of the coupling

Hamiltonian with the classical field. We obviously retrieve in this way the formalism of Chapter 2.

4.4.3 Non-resonant coupling and single-atom index effects

Remarkable phenomena, which can be interpreted as single-atom refractive index effects, are alsoexpected when the atom and the field are not at resonance. Exact expressions of the atom–fieldeigenstates and energies are given by eqns. (4.101) and (4.102) for an arbitrary detuning ∆c. Ratherthan discussing these general expressions, we focus here on the large detuning case, for which aperturbative treatment leads to simple formulas with a clear physical interpretation. A mere inspectionof Fig. 4.3 shows that for |∆c| Ωn the dressed levels tend towards the uncoupled ones. When∆c → +∞, we deduce from eqn. (4.100) that θn → 0, entailing that |+, n〉 → |e, n〉 and |−, n〉 →−i |g, n+ 1〉 (eqn. 4.102). These conclusions are reversed for large negative detunings: when θn → π,|+, n〉 → i |g, n+ 1〉 and |−, n〉 → |e, n〉.

Since the dressed states almost coincide with the uncoupled ones, the probability of an atomictransition accompanied by photon absorption or emission is negligible, as expected from an energyconservation argument in this off-resonant situation. The energies of the dressed states are howeverslightly different from the uncoupled ones, as seen on Fig. 4.3. These energies can be expressed


by expanding eqn. (4.101) to lowest order in powers of the small parameter Ωn/∆c (for the sake ofdefiniteness, we assume from now on ∆c > 0):

E±n = (n+ 1/2) hωc ± h(

∆c

2+

Ω2n

4∆c

). (4.108)

Figure 4.4 shows the effect of this energy shift on two adjacent doublets of the atom–field system. Ineach doublet the level spacing is slightly increased, the effect being larger in the upper one. Morequantitatively, the shifts ∆e,n and ∆g,n of the states |e, n〉 and |g, n〉 are:

∆e,n = h(n+ 1)s0 ; ∆g,n = −hns0 , (4.109)

with:

s0 =Ω2

0

4∆c. (4.110)

These shifts, which affect globally the atom–field states, can be given complementary physical interpre-tations, depending upon which of the two interacting subsystems we are observing. They correspondto reciprocal changes of the atomic transition and field mode frequencies.

The atomic transition frequency measured in a field containing n photons is equal to 1/h timesthe energy splitting between levels |g, n〉 and |e, n〉 (Fig. 4.4). It is thus shifted from its free-spacevalue by:

δωeg = (2n+ 1)s0 . (4.111)

This shift is the sum of two contributions. The first one, 2ns0, is proportional to the photon numbern. It corresponds to the so-called light shift. The second contribution in eqn. (4.111), equivalent tothe light shift produced by ‘half a photon’, is observed in the vacuum. It is a Lamb shift effect, dueto the coupling of the excited atomic level to the vacuum fluctuations in the mode.

The Lamb shift effect is usually very weak and can be observed only in the CQED context. Thelight shifts in intense laser fields are much larger. They are the basis for the so-called ‘dipole force’trapping of cold atoms. An intense laser beam, detuned on the red of the atomic resonance, is tightlyfocused on cold atoms in their ground state |g〉. The light shift of this level is negative. The energyis thus minimum at the maximum of the field intensity. The light shift is an effective conservativepotential for the atomic motion. In simple terms, the atoms in the ground state are attracted andtrapped near a field intensity maximum, hence at the focal spot of the laser beam. The trappingdepth can reach rather high values, well in the mK range. Dipole traps are widely used for atomicsample manipulations and Bose Einstein condensation. Note that the phenomenon is reversed for ablue-detuned laser, the atoms being then expelled from the high-field regions. Proper combinations ofred and blue detuned fields can be used to tailor interesting potentials, in particular in the immediatevicinity of very thin optical fibres.

Optical lattices are a variant. Atoms in a three-dimensional standing wave light field experiencea periodic trapping potential. The atoms thus behave as particles in solid state, also subjected to a3d periodic potential. In this respect, optical lattices can be viewed as a quantum simulator for theexploration of complex quantum phenomena.

The dressed levels shifts, viewed from the field perspective, correspond to a change of the fieldmode frequency. When an atom in level |e〉 interacts non-resonantly with the field, the |e, n〉 levelscorresponding to different n values are equidistant, with a spacing h(ωc + s0) (Fig. 4.4). This meansthat the field frequency is offset from ωc by:

δeωc = s0 . (4.112)

For an atom in level |g〉, the |g, n〉 to |g, n+ 1〉 spacings are h(ωc − s0), independent of n. The modefrequency is thus displaced by an opposite amount:

δgωc = −δeωc = −s0 . (4.113)


These shifts have a transparent interpretation in terms of refractive index. The atom, which cannotabsorb or emit light in the detuned mode, behaves as a piece of transparent dielectric. It plays thesame role as the dielectric plungers used to tune cavities in microwave equipment. The atomic indexchanges the mode frequency. This effect is dispersive (frequency dependent) and s0 has the same signas ∆c.

Standard optical index of refraction is certainly rooted in these shift effects. It is neverthelessa much more complex phenomenon, since it involves dense matter (interatomic interactions are nonnegligible) and since the atoms can barely be considered as two-level systems. However, again, theCQED context allows one to observe and use these dispersive effects.

4.4.4 Two applications

Autler-Townes splitting

We return here to the discussion of Section 2.6.3. We consider a three-level system, with two long-livedlevels |g〉 and |f〉, both connected to an excited state |e〉 by two electric dipole transitions with quitedifferent frequencies. The |g〉 → |e〉 transition is addressed by a strong resonant laser. A much weakerprobe beam explores the spectrum of the |f〉 → |e〉 transition dressed by the intense laser. Obviously,the |f〉 level is not affected by the dressing laser (which is far from resonance with the |f〉 → |e〉transition).

The strong dressing laser is in a coherent state with a large average photon number. As shownabove, it can be assimilated to some extent to a Fock state |n〉. The uncoupled levels are thus replacedby the dressed states |±, n〉, superpositions with equal weight of |e, n〉 and |g, n+ 1〉, separated inenergy by hΩn. The level |f, n〉 (atom in level |f〉 with n photons in the dressing laser) is coupledto the level |e, n〉 by the probe laser (which can change the atomic level, but not the photon numberin the dressing laser). Hence, both the dressed states |±, n〉 are coupled to |f, n〉 by the probe laser(albeit with a matrix element reduced by a factor 1/

√2).

The spectrum of the |f〉 → |e〉 transition, instead of a single line at the naked atomic frequency,is thus made of two (ideally infinitely narrow) lines separated by the dressing Rabi frequency Ωn andcentered on the naked atomic frequency. This is the Autler-Townes splitting, which can be viewedas a direct spectral counterpart of the temporal Rabi oscillation. Of course, this spectrum can becomputed from the solution of the optical Bloch equations, as mentioned in Chapter 2. However, thedressed atom approach in much simpler.

We predict here infinitely narrow lines, since we have not included any relaxation process. A propertreatment of spontaneous emission in the dressed atom picture will be outlined in the next paragraph.We leave to the reader as an exercise to extend it to the present situation and to show that the twocomponents of the Autler-Townes splitting are equally widened by spontaneous emission.

Mollow fluorescence triplet

We are interested here in the spectrum of the fluorescence emitted by the atom under a strong laserdrive, observed for instance by collecting the light emitted in a direction different from that of thedressing mode.

We thus clearly need to incorporate spontaneous emission in the picture. As for a free atom, thejump operator in the Lindblad equation must be

√Γσ−, where Γ is the spontaneous emission rate and

σ− = |g〉〈e|, which transforms |e, n〉 into |g, n〉. The atomic lowering operator thus acts on the dressedstates as σ− |±, n〉 = |g, n〉. Since both dressed levels |±, n− 1〉 involve the uncoupled level |g, n〉,fluorescence-induced quantum jumps may occur on all four transitions between |±, n〉 and |±, n− 1〉.The transitions between the two + and the two − levels are at the naked atomic frequency ωeg. The‘crossed’ transitions |±, n〉 → |∓, n− 1〉 are at ωeg ± Ωn (we can neglect the variation of the Rabifrequency between the two adjacent multiplicities).


The dressed atom spontaneous emission can thus be viewed as a sequential cascade through theladder of dressed states (which can be viewed as infinite since the photon number dispersion is muchlarger than unity in a strong coherent field and also since the laser source renews constantly thephotons in the dressing mode as they are scattered by the atom).

The spectrum of fluorescence is thus made of three lines (hence the name Mollow triplet), one atthe naked frequency and two sidebands. It is quite clear from the above discussion that the centralline is about twice more intense than the sidebands. A detailed calculation of the weights and widthsof the three Mollow triplet lines can be performed based on the Lindblad equations. We refer theinterested reader to the Cohen-Tannoudji’s textbook Photons and atoms.

Another effect can be readily predicted from the dressed atom picture. Once the atom has emitteda photon on the lower sideband, it is left for sure in the |+, n− 1〉 level. From there, the next photoncan never be emitted in the lower sideband again since this sideband originates from the − dressedstates. The photons in the two sidebands have thus a negative time correlation (anti-bunching).This effect is very easily predicted in the dressed atom picture and can be exactly computed. It isconsiderably less evident in the optical Bloch equations frame.

We have discussed in this section two applications of the dressed atom picture. There are manyothers, and this approach is most certainly the most fruitful one in quantum optics.

4.5 Cavity Quantum electrodynamics

The final Section of this Chapter deals with the Jaynes and Cummings model (1963), a fair andprecise description of a Cavity Quatum Electrodynamics situation, when an atom is placed insidea high-quality nearly resonant cavity. The coherent interaction of the atom with the field can thenoverwhelm the dissipative processes. This very active domain is interesting, for its applications, butalso because it leads to direct illustrations of the quantum postulates and to new light shed onto thequantum nature of the fields.

The large number of experimental figures and of schemes and drawings makes it easier to give thelectures as a slides presentation. These lectures notes will thus be replaced, for this final section, bythe handouts of the slides.

m2 atoms and photons complete lecture notes

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